Backups failing for all databases
I should have written this long ago, but better late than never.

How did I resolve the issue? Each backup attempt created a dump file which I analyzed using WinDbg. These files indicated that the error originated in the sqlmin.dll assembly. I did a search for hotfixes for that assembly, found some, applied them, and after a reboot, the backups started working again.

For anyone who comes across weird SQL Server bugs like this, try looking at the dump files for clues.