Every secret agency gets a secret base. The agency wouldn't be secret if it didn't had a secret base and even some old unconditioned administrative building is a secret base if nobody knows what happens there, there is a lot cover around it and such. It just so happens that it is an underground facility. Not the biggest one to be honest. Hangar is something about the size to fit an Osprey (which is one of the biggest "ships" in terms of a lenght in any direction that X-COM can get, probably), while rest of the facillities aren't too big either.
The best thing we have now is to think around those limitations by doing something fun (atleast for me), which is "plotting the story".
Its not like UN has that little faith in X-COM, even from the start. At the point where we take command over the organisation and forward, Council has basically only two plans for dealing with alien threat. One is X-COM and other MiB. Both with very different approach. X-COM had many predecessors who showed some progress, like Xenonauts and "Kira-kuy" (something like that with the name, I didn't recently play the mod).
All they require are proofs and to gain those proofs, X-COM had to have good enough conditions to start. One, fairly small underground base, few engineers and scientists aren't much, really. Every country in UN (and basically every country in the world, since X-COM can get base in every part of the world which might not be "cannonicaly" up to player's choice, but a predefined, while player gets to choose, or the player's character, the "Commander" might himself look forward to create such organisations and intentionally influence and "nudge" the things to create first base in a place he choosed) could realistically have such underground base, X-COM don't get much from the start, really.
Lack of weaponry on the start is already explained with cross-border tentions that not many goverments would like forgein agents bringing up (depending on each country politics) strictly non-civilian weaponry. Trough time, evidence gathered and such, the Council agrees to give X-COM bigger freedom, with explosives and machinegun licenses.
Seems reasonable. How one could think that a president/dictator/whatever of some country would react to being forced to allow several armed to the teeth agents to operate on his territory, killing and arresting "suspects" as they please? Then basically kidnapping those arrested and transporting them to the other part of the world of course.
I wouldn't really call the budget that small, since X-COM in the beginning is very small itself, two field agents, bunch of scientists and engineers, not much really.