The server this site is hosted on went down.
It wasn't that unexpected (it went down twice unexpectedly), but it could have picked some other time than 11:00 am on a Sunday to go down.
Anyway, preparations where made for such an emergency (after the last debacle) and the transition of sites to a new server went, for the most part, smoothly. Started the backup. Went back to sleep for an hour. Checked, went back to sleep for another half hour. Sites finally restored on the other server, schlep over the configurations and everything was pretty much up and running.
A small digression. On each server, I have the Apache configuration file broken up into pieces, which are then patched together to make the final configuration file—there's a section for the base configuration, another section listing the IP addresses and then sections for websites and websites with a secure certificate. I just had to patch these pieces into the configuration file on the other server.
The only small problem were a few scripts that had some very old paths hardcoded in that had to be modified (could have been fixed with a symlink, but that would have broken about a hundred other sites that used said symlink to work—but it was only about six sites that needed fixing).
Now we just need to retrieve the server and see what failed (I'm guessing the harddrive, given how it was serving up webpages quite slow, yet there was no load and the network was not saturated at all).
I'm hoping tomorrow is quiet.