I noticed that under System Settings the Debug Mode was set to Development. Changing it to Production made the errors go away and now I get the happy "database backed up successfully" message.
But, new problem: when I FTP into the Directory set in the Directory Path For Backups field, I find no file! This is true on both sites.
And, isn't "777" a rather un-secure permissions setting? I mean, that basically allows the "public" to go in there and muck about, right? In fact, on the Host I'm using, I get Internal Server errors if the directorys are set to anything but 755.