During a database backup last night, the filesystem where we store temporary (non-critical) database files got borked somehow. For some reason, the database's cope, likely recreating the temporary files over and over again without anywhere to store and reuse them, caused it to get really slow. It took us awhile to figure out that was the cause and it was fixed by literally turning it on and off again.
Thank you to everyone that reported it, and sorry for the downtime and the verbose error messages. (It's not wholly intentional to show you the sausage making but it helps us diagnose issues easier until we create some kind of error digest system.)