A character waunders in and orders "a steak" in your basic fantasy world. Â It is not only a question of would they have one, but the question becomes COULD they have one.
Depends how far back you are going.
In the wild west it wouldn't have been unreasonable to keep a side of beef around for a few days, if not longer. Â Dry Aging not only improves the taste of beef, it preserves it. So keeping beef in a very cool somewhat dry place enhances the flavor and keeps it. When you have your meat in camp, the hard part is done, but there are still some critical steps before it gets to the freezer. The most important thing is to keep the meat cool and dry and allow air to circulate around it. You want a tough crust to form around the outside of the meat. This crust will keep bacteria out while the meat ages. During the aging process, the muscle proteins begin to break down, and the meat becomes more tender and flavorful. The amount of time that you age meat depends on the type of animal, the hanging temperature, and the humidity level. Hanging time ranges from one day to a couple weeks. In fact most butchers won't eat "fresh meat", it is really that much better after hanging around for a while. Â
In the wild west, and times past there were a couple of other options.
1. Brine pack in rocksalt was very poplar for meats and fish.
2. Sulphering was used mostly with fruit crops.
3. Drying, virtually everything could be had in a dried format.
4. Honey pack was used quite a bit with fruits and some meats.
5. Lard pack was very popular with meats particularly beef and pork. Cooked meat was casked and molton lard was poured over it.
6. Canning and Jarring were very popular by the 1860s.
7. Ice boxes were pretty well unknown out west though some of the more upscale individuals and establishments did indeed have them. Ice was harvested in the winter from regional lakes with saws and stored in sawdust until summer.
In the wild west, of course there is the option of just placing a telegraph order and having frozen goodies delivered by train, if you had enough cash of course.
Going further back there's a few ways of keeping stuff cool and fresh. A cellar will almost always be cold in the days before insulation. A room with water flowing through it (a room built over and open to a stream for example) will also stay cold.
A good option is keeping it as an animal as long as possible - chickens would be killed on the day they were to be eaten, and larger animals like pigs would be turned into lots of preserved produce (like bacon and smoked sausages). Keeping the stuff hanging in your smokehouse would preserve it longer. But meat in the middle ages was frequently slightly rotten, hence the popularity of spices when they became available.
In games with magic, magic might be used to preserve the food. Of course we have to ask is how common is magic? Â Is it cost effective to pay a mage once a day, once a week, or once a year (or for some permanent spell) to preseve the food? And are there enough mages to support this? If mages are one in a million, then only Kings/ Queens will have access to them. Â If there is one around every corner, making all those magic items that PCs seem to pick up, then it would be possible.
At the end of the day though, it's probably fair to say that inns are a fairly modern invention in a psuedo euro medival fantasy. You don't have to go too far back to a time when people don't have the spare money to eat out, you don't have to go that far back to a time when most people didn't even have money. I suspect the only places like inns were coaching houses that could rely on a constant supply of travellers. I can't imagine they were offered steak or the such like, they probably could have whatever they wanted to eat as long as it was stew. But by modern remember this towards the end of the middle ages, early renaissance (Italian/ English/ Germanic) . There is a pub in Dublin, Ireland--the Brazen Head--that's been operating on the same site for something like eight hundred years. And there are others of similar pedigrees.
Of course, in other parts of the world this was different. There is a restaurant in Kaifeng, China, called Ma Yu Ching's Bucket Chicken House, which supposedly opened it's doors for the first time in A.D. 1153. (I wonder if they've still got all their old cookbooks, tools, and whatnot.) And this location is not the oldest mentioned in the record. Â Of course China never had a medieval period, and has been quite urbane for 4000 years.