It seems that someone at the host meddled with our droplet. I went in and re-applied fixes, so we are again stable. (We do however, still reboot every day at midnight GMT.)
I also updated some other system components in the process, and re-fixed FTP.
My apologies for the downtime, but I had nothing to do with whatever caused this, and it deeply troubles me that someone not familiar with the server was fekking with it.