One of ours broke for no good reason; Some kind of SQL engine screwup....
I tried various self-repair tools, googled some of the "hey try this" fixes and failed. It wouldn't even restore from a partial backup properly (possibly because it's a server role now, rather than a standalone product) so I just installed WSUS on another server and changed the WSUS source to the new server in AD. All machines were happy within a couple of days or so, since WSUS really is one of the easiest, hand-holdiest things I've had to configure in a while.
One day I'll rebuild the original WSUS server but it's not doing any harm to the other roles so I'm sticking to the "ain't broke, won't fix it" principle.
Is there a way to back up all the settings (computer groups, etc..), but install a new database?
It depends how it was initially configured. If it was set to use AD groups to sort/filter updates for connected clients then there's nothing to backup (unless your AD groups are shot, in which case you have far bigger problems than WSUS). If it was managed from within the WSUS web-interface then I can't remember how to help you, but I don't think WSUS 3.0 allows you to do this (I'm probably thinking back at least five years now).