Problems with a database? Almost never. The only issues I typically run into are when a backup fails. Our environment isn't monitored by the engineers even though we have two tools to do so. We frequently run out of free space. The other issue is when Legato has a lock on the backup file - always in SQL 2000 so it is becoming less frequent.
All issues are handled remotely.
I am not only our SQL DBA, though, I also handle about 20 different systems as the only analyst for them. So I get paged or called far more often for issues with systems than databases. SQL Server does not typically stop running, apps do.