Or is it an unfeasible idea?
If you're talking $ you probably know already that price of electricity is your worst enemy, and I see you're from the UK (??) auch
As for the rest, soundproofing shouldn't be an issue if you plan on building a bunker, minimal insulation between the concrete pad and the floor with rock wool will deal with it. The problem would be the airflow, pushing hot air outside from underground is simple, pumping fresh air might get tricky depends on the scale you have in mind. There are a few members who have insulated their garage, a room, a basement but I don't remember somebody putting them inside a bunker.
You might get a few hints from
Steamtyme's topic.