The creatures eat rocks. The excreta from this is fertile loam that is ideally suited to farming.

They can be trained to eat from rock formations in such a way as to create walls and fortification.

If they eat a metal ore, the metal part is not digested and can easily be separated from the loam to produce pure gold/silver/iron etc.

A problem is of course that in the wild they take chunks out of random mountains or even stone castles if not carefully monitored. Thus the landscape will tend to become flatter over many millennia.

There may be some huge underground caverns and tunnels where they have made a den.

Usually such creatures are at least partly based in magic and therefore don't need the usual degree of physical sustenance that would be expected. If you want to ignore this and go for purely biological solutions I suggest referencing Australia's saltwater crocodile which can and will lay up for months hibernating between meals; your dragons don't need to eat often just heavily when they get the chance.

Like whales, your dragons feed on krill-like insects

After all, whales are also "castle-sized", and they manage to find enough food in a sustainable way.

So, your world contains legions of very small flying insects, swarming quite passively. Your dragons have found a way to harvest a lot of them efficiently (a blue whale can eat 4 tons daily). This may add an unwanted constraint to their design, though.

These krill-like insects feed on phytoplankton-like micro-organisms carried by the wind, reproducing by fission, and photosynthetics.

Your dragons may seem a bit less cool and dangerous this way... But nothing stops you from also having a few "regular" carnivorous beasts, at the top of the food chain. Those would be very rare and deadly.

When you keep the environment temperature constant, you can make your beasts ectotherms. That would reduce their need for food, since their metabolism would not need to maintain a constant body temperature. They could much more easily survive longer periods of time without food.

If you add a rich vegetation that is capable of maintaining large herbivores (like the brontosaurus), you also give them sufficient food to sustain themselves.

In this article you will find a good comparison of the metabolisms of endotherms and ectotherms (and also some further references).

Citing from there:

Endotherms tend to have basal high metabolic rates and high energy needs, thanks to their maintenance of a constant body temperature. Ectotherms of similar size tend to have much lower standard metabolic rates and energy requirements, sometimes 10%, percent or less of those of comparable endotherms

Also, according to the link, the larger size works in favor of your dragon's metabolic needs, since it needs less food per mass:

Which one has a higher basal metabolic rate: a mouse or an elephant? If we look at the metabolic rate of the entire organism, the elephant is going to win – there is way more metabolizing tissue in an elephant than in a mouse. If we look at per-mass metabolic rate, however, the situation flips. A gram of mouse tissue metabolizes more than 10 times faster than a gram of elephant tissue!

Maybe they eat uranium/thorium, have stomachs that enrich the fissile elements and extract energy from nuclear fission. Or they have a breeder reactor on their guts. You will need magic to save their cells from the hot neutrons and gammarays.

Dragons eat outside the terrestrial biosphere

House sized dragons don't need to eat on the surface of the Earth. Perhaps they just like to rest here. Take the idea that they eat plankton to the next level: they eat something that exists outside of the food chain interactions of the planet's surface.

The most obvious answer is the ocean. Lets say your planet does not have any whales. Everything that a couple million whales eat on Earth is instead eaten by a few dozen mountainous dragons.

Another good answer is the underground. There is an immense biosphere of mostly microbes under the surface. Your dragons are tremendous diggers who "eat" rock and sift out the biological materials.

The last and most fun option is leaving the planet entirely. Perhaps there is a forested moon that they travel to. Perhaps there are space plankton.

Okay, first of all, we'll need to figure out how much food it really needs. Let's say your dragon is the size of an Amphicoelias fragillimus, a real castle-sized land animal, which is estimated to have weighed 120,000 kilograms. Assuming a reptilian physiology, the dragon would need to consume 1.2 million kilocalories a day. Damn.

That's what the formula coughed up at least; however, I suspect the daily energetic requirement would be lower, since it's the same as a blue whale's, which is both heavier and an endothermic mammals. Let's call it a million.

To make this more plausible, I recommend the following planetary conditions:
* High atmospheric oxygen levels, ideally about 30%
* A dense atmosphere
* High global energy availability to maintain many-leveled food chains with broad bases

Low gravity isn't very important, since it makes the dense atmosphere less likely, which you need more. Now, to properly answer your question:

First idea; aeroplankton. Earth has aeroplankton in real life - viruses, bacteria, fungi, protists, algae, spores, pollen and seeds, all carried on the wind. However, there's not that much of it, it's not very dense, and a lot of it is too high up to be harvested by your dragon. Given a dense atmosphere, it's plausible if not likely that these problems wouldn't be present.

Alternatively, you could take inspiration from the aforementioned Amphicoelias, which was a "veggiesaur". It was essentially the size of a castle, as you can see above, and it got by eating just plants. Specifically, it stripped the branches and bark of large trees, probably clearing whole forests like modern elephants.

Given that sauropods of such size could've existed, castle-size, even on land isn't that far fetched. However, I doubt your dragon could fly except in a very dense atmosphere. You could decrease its caloric needs by giving it an even slower metabolism (which would make it sluggish) or having it hibernate for long periods.

Dragons are beach scavengers in the model of Tyrannosaurus - or the condor.

Here is my answer to this question.
Mythical dragon diet

http://www.montereycountyweekly.com/news/local_news/condors-make-a-meal-of-a-beached-gray-whale/article_aabcf553-a462-502d-8809-93225ba069f1.html

The California condor is (!yay) the largest North American bird. After the Pleistocene megafauna died out, condor populations persisted in California where coastal populations could sustain themselves largely on beached whales. Among other things, the crash of the whale population midcentury was really hard on the condors.

A dragon could be an unparalleled coastal scavenger in the mode of the condor. Really large animals are thrown up by the sea. Condors are big and tough and can keep a carcass for themselves, driving off everything except a bear or an eagle. Nothing much is going to drive a dragon away from its carcass except another dragon - or maybe some sort of sea monster Liopleurodon dragging itself out of the sea to feed. In a world with dragons, things weirder than whales might wash up. Some things might still have some life in them and resist being eaten by dragons.

My piece of advice is one related to preventing them from being murdered by our good friend the square-cube law. You will want to probably give them a really low metabolism (which also happens to reduce food requirements), and make them ectothermic. That way they won't absolutely burn up.

Or you could make them endothermic but really resistant to heat and have their body temperature be constantly insanely hot. Additionally, a castle-sized dragon, without the aid of magical physics, would be unable to fly. But that is not to say they shouldn't keep their wings, it's just that wings would serve an entirely new purpose. They would be used to dissipate heat into the environment if they were endothermic.

In fact, what if we took this further and said that they don't breathe fire but really hot air that dries out and ignites things. If the temperature was increased enough you could have them melt the incoming armies using this superheated air. This can be used to damage foes, but also would be an excellent way for them to cool down during a battle.

The other issue we face is that they are gonna have a problem with their insane weight. They are gonna want muscles that are vastly stronger than what a human can come up with, along with a really strong skeleton and definitely have the whole scales like armor thing. That will come in handy.

These castle sized dragons would also have the unfortunate side effect of appearing slow, but in actuality, they could be moving at 80 kilometers per hour. You would want them to have flesh that is much less dense than that of a human, aside from the bones and skin of course. Such a dragon would be able to roar incredibly loudly, possibly such that could be used as a tactic to frighten, disorient, or burst the eardrums of enemies.

My main advice for keeping them well fed should be that they go into dormancy most of the time. But I'm not talking about simply sleeping, I'm thinking they should be almost dead and take several years to come out of it. When they awoke from hibernation they would want to target human settlements for one specific reason, livestock. Assuming they were carnivores than they would very likely want to go to places with a very large volume of meat before their next hibernation. I'm not a scientist and some of this could be wrong, but if you incorporated a little physics into this these creatures could be very interesting indeed.

Add O2 to the atmosphere

There was a time on Earth, the Carboniferous period, when ferns grew to the size of a tree, scorpions to the size of a small dog and dragonflies reached the wingspan of a hawk. Plants covered the planet due to warm and humid conditions around the globe, releasing huge amounts of oxygen to the atmosphere. This "surplus" oxygen allowed insects to grow big enough to sustain a dragon diet.

... and control metabolic demands

Moreover, a warm weather helps to keep metabolic demands at lower rates, so ectotherms can thrive.

It also helps if your dragons are not active the whole year, make them hibernate or go through a dormant state for long periods.

Ectotherms have lower metabolic requirements but they do not like cold weather (no cocodriles in the North Pole), however, aquatic ectotherms seem to do better in a cold climate (salmons, etc.)

To answer your question, I think it is important to understand what the foundation of your new world is. What @Chasly said about rocks is fun and all, but this can only be believable in a world entirely different than our own. As an example, you could change the fundamental laws of the universe (electromagnetism, strong and weak nuclear forces, and gravity). Perhaps that would be too complex, but there needs to be a reasonable way that consumption of rocks could provide an adequate amount of energy, especially considering the fact that rocks don't contain a lot of readily accessible energy for other organic forms of life. Any way of harvesting that energy would likely release too much energy as a result.

Is it important to you that the dragon eats something small? Is it important to you that you completely remove the normal circle of life that we are accustomed to? Can we not have a larger planet? This would make it easier to stay consistent with the rest of your world because you could take what we already know and just make the potential for everything to grow bigger because there are more resources. That may also necessitate that your characters (if any) would also be bigger, and be in a world with everything relatively bigger but not bigger relative to themselves. That may take away the appeal of larger creatures because their power is no longer relevant in a world of relatively more powerful creatures (everything is relative). Maybe in order to maintain the majesty of large dragons you make it a world that visitors travel to, perhaps on accident. Then you could have this world of large creatures that remains impressive (this is sounding a little like King Kong, which may be a good thing?).

No matter what you choose, there needs to be consistency. If what the dragons consume is important, maybe build a new world around that. If something else is more important, build your world around that. State the assumptions of your world, and then we can productively and effectively design a solution that works in your world. If there is no consistency, nor struggle in the world or in its design, then it becomes boring. Ultimate power is not exciting (just ask Superman, who seemed to intentionally develop an allergy to a shiny green rock).

They could also be similar to cecropia moths (naturally not as fragile though). Cecropia moth caterpillars eat and when they grow into the adult moth they no longer need to eat (they rely on fat reserves). So it could be that the dragons eat while they are smaller and as they get bigger just simply don't need to eat at all, or very rarely.

https://nhpbs.org/natureworks/cecropiamoth.htm

just do like with normal sized animals :

• have some herbivorous/omnivorous eat forests


• have carnivorous eat the herbivore ones


• then the plants will grow much bigger thanks to having so much ressources from the dragons' corpses

and that close the loop

the only difference is that they're bigger

Depends on how "hard" you want your fiction. Right now the top answer suggests having your creatures eat rock, which is a cool idea but doesn't make a whole lot of sense from a biological point of view, not if we assume your creatures are carbon based like life on Earth is.

So, I'm just going to throw a few more options with varying degrees of "hardness" and plausibility. But first, a premise:

I'm assuming that your world will be Earth-like, with comparable atmosphere and temperature. I'm also going to assume that you wish to handwave the other classic problems one runs into when designing giant creatures, like how do they breathe or how do they avoid collapsing under their own weight.

With that said,

1. Have them rely on symbiotic organisms that can metabolize stuff that a normal animal wouldn't be able to draw sustenance from. Photosynthetic organisms living in or on their skin could provide a lot of energy, as would a variety of specialized bacteria living inside them and able to process dangerous chemical substances that normally need to be disposed of, like the colonies of bacteria living in abyssal tube worms. This would even make the lithovore suggestion more plausible.


2. Have them live in the water and rely on filter feeding or similar behaviour. Just like blue wales, the largest animals that ever lived. If you want them to be terrestrial, have them have access to similarly abundant sources of food in the form of small, gregarious and numerous animals. Just keep in mind that this will necessitate a very different environment from Earth in order to work, probably a hypercharged biosphere, with access to a lot more energy than what is found on Earth. Some kind of super-earth planet might do the trick.


3. Have them be plant eaters. Boring but practical. As the biggest land animals that ever lived (sauropods) show, some truly impressive sizes are possible for a sufficiently specialized plant eater. And of course where there are large preys there are large predators, though to have them in any large number or to make them reach as impressive sizes as their preys you need again a hypercharged biosphere.

But how am I supposed to keep them well fed?

I would combine three things.

1. Your house sized creatures (dragons, etc.) sleep a lot.
   
   Say three of a farmer's fat cows will sustain them for a week or two. The first week they could be in a semi-coma - which is why it is important to keeps their lairs location a secret and hard to get to for humans. Lots of story building ideas here plus it matches existing aspects of mythical Earth dragons.




2. The prefer to sleep in caves near volcanoes, or near hot springs, or just caves deep enough to be naturally hot. Lets them sleep longer because they burn less energy. I wouldn't go into whether they are warm/cold blooded unless you have a plot that relies on it... after all dragons are generally considered to be 'part magical' creatures. Bringing us to point three:




3. The magic which helps those hose sized things fly, also sustains them in other ways. The explanation can be left to the reader which can enable you to gloss over the specifics of the first and second point.

Dragon has an ability to open portals to more fertile lands. Then he feeds in there and comes back.

If you are talking conventional dragons -- large reptiloid appearance, capable of flight, then you either need magic, or you have some serious world building problems to tackle.

The closest analogue I can are the Pterosaurs. The largest of those had a wing span of about 36 feet. About the same as a Cessna 172. But they were flimsy beasts. One 18 foot speciman was estimated to have a weight of about 55 pounds. I would expect a 36 footer to have a mass between 200 and 400 lbs.

A denser atmosphere is clearly a requirement, and a lighter gravity would be beneficial. We could probably raise the atmospheric density by a factor of 4-6 and still be able to see. (Too much air and the surface is as gloomy as Venus.) To get lighter gravity, we need a planet that is larger, but less dense. This would mean a smaller core, with less iron.

Problem: The molten iron core generates the magnetic field that keeps the solar wind from ripping off the atmosphere. (One of the reasons proposed for Mars having to little air.)

Ok. Spin the planet faster. Let's give it on rotation per hour. That would give a small core a lot more speed. It would also make the planet much flatter. (For an extreme of this, see Hal Clement's novel "Mission of Gravity" Mesklin was a rocky planet with Jovian mass that spun at 1 rev/10 minutes. Resembled a fried egg.)

With a faster spin, and a denser, deeper atmosphere thunderstorms would be wild! Your dragons may have developed intelligence just to cope with the weather.

Despite these changes, I don't think they will be that tough to kill.