Now that would be a great location to put a hotel or even a home. But which mountain?
Seriously, if you type ‘house built in a mountain’ (or even hotel) into your search engine and click on images, such places really do exist – houses, hotels, hydro-electric power stations, all kinds of buildings that have been really built inside mountains! However, I do not think this is what our speaker means.
I think we can assume that they are talking about some kind of accommodation that they plan to stay in, but unless they are actually planning to relax in the examples listed in the previous example, they are more likely to be staying in a place that happens to be located not in one particular mountain, but in among a group of mountains that the listener knows: The Tatras, the Alps, the Dolomites, for example. (Note that when we talk about groups of mountains, we make the name plural if the word 'mountains' is not used as part of its name: The Pyrenees (NOT The Pyrenee); the Atlas Mountains (NOT The Atlas Mountain); The Rocky Mountains (or The Rockies).
So our speaker should perhaps say one of the following examples: The place where we’re staying is located in the mountains (within a particular group of mountains known to the listener, NOT one particular mountain), or the place where we’re staying is located on the mountain (on top of, or placed somewhere on, a mountain that is known to the listener as the definite article ‘the’ is being used).
And to make clear one other preposition problem: we can say I'm going to the mountains for my holidays (destination), I'm going in the mountains to pick some mushrooms (to do an activity like walking or simply sightseeing in the area where I am already located).