Now, hear me out. But, I think that Vault 111 might actually be based on a brewery.
Boston actually has a very long history of breweries. In fact, prior to the in 1920 Prohibition law passed in US Congress and ratified by 2/3'rds of the US States at the time of its passing, it was known as basically the US beer capital. Where you could go and try out various beers that weren't available anywhere else (until the creation of refrigeration).
Breweries were in fact built atop or even into hills, and often had big deep basemants. These basemants were used to store and keep the beer cold before the advent of refrigeration. I'm willing to bet the cryostasis or whatever form of stasis they are using too keep the PC and others alive (as long as the system was working and maintained), is on the lower floors of the vault.
Beer bottles are pressurized, much like how it appears the Vault might be pressurized from the outside world. And when the elevator door opens too the outside, it sounds a lot like a beer bottle opening up. It even looks like a beer cap that hasn't been pressed yet!
Speculation? Yes. But if you think about it, both preserving beer and preserving people through cryostasis, would require cold temperatures. And when you consider Boston's history as being early America's Beer capital, and as far as we know it, there hasn't been a mention of Boston's long brewery tradition.