after about 11 hours of downtime yesterday.
Apparently, the /home partition of the server went bad yesterday afternoon. According to Site5, "The server has experienced a failure with the primary hard drive and will need to be restored to a new machine."
It seems we're on the new machine and things seem to be working.
(Now I'm no expert on hosting, but it seems that this should have been done a month or so back when the databases kept failing almost daily. It seemed back then that the server had reached its limit. But again, I'm no expert.)