Refuge?
Or prison?
I think you need people to maintain a facility like that. And in the event of a catastrophe I have a feeling the staff may not be as conscious of their place in life. Downstairs might turn on upstairs and the busboy rebellion could get ugly.
1. Given the alternative, I'd rather be a survivor.
2. A "luxury cave" doesn't sound all that bad!
3. I expect that most of the billionaires would be old, fragile coots (often with nubile young wives). In view of that, I wonder how long it would take until the host of young, healthy maintenance workers, well-muscled security forces, and discontented kitchen workers necessary to run an operation of this kind simply seize the opportunity and take over.
The billionaires would thus soon find themselves back on the outside, along with the other scavengers.
Regards,