More information from the CEO of Autonomica, the organisation that runs some (all?) of the .se infrastructure.
Autonomica == Netnod, which runs most of Sweden's Internet infrastructure as well as i.root-servers.net (one of the 13 root DNS servers).
The Internet still worked, however domain names didn't. You can use the IP addresses directly still, ja? Also, alternative DNS servers could be used still, no?
I certainly hope there's a more robust process for ".com".