The server has been unavailable to most users from guesstimately yesterday evening until 13:30 (CET) today. Services were suddenly being denied access to various system locations (such as /tmp and /dev/null, which are rather vital).
While the exact cause is still being investigated, the most likely culprit is Gentoos package management software (portage). Whether it’s a general bug or "just" an error in one of the package ebuilds I do not know, but I’ll certainly try to find out.
I guess this means it’s time to go for a proper ACL system and Tripwire (or some other integrity checking tool).