We had an intermittent outage, but now the site is back up.
The outage was caused by an upgrade to the MySQL client libraries, which made them not want to talk to the server anymore. The error code was ERROR 2027 (HY000): Malformed packet. The solution is described on StackOverflow.
The resolution was to upgrade the DB password on the server to a new algorithm, which was a fix done in three minutes.
The hard parts were getting home, looking through all the logs, and finding the error :)