What happens to unpowered living quarters and storage rooms?

Solution 1:

If the rooms are unoccupied, nothing happens. You gain nothing from populating Storage Rooms or Living Quarters, other than potentially affecting Vault Happiness. (Some Dwellerss can gain happiness from being assigned to rooms). If the power goes out, they'll just wander around the room doing nothing (unless you are trying to get dwellers to hook up - they won't when they power goes out).

Obviously, for rooms like training rooms, or the armouries, etc, those rooms will stop production, but it won't cause any real issues. The radio room will either stop broadcasting, and it can affect Vault-wide happiness.

Other than that, losing powers to these rooms will not affect your vault to any real degree. You won't lose storage or Vault Capacity or anything like that.