Hello everyone,
An update: we've started a backup server and updated the DNS records to point to it. As Marius and Mike have already seen, connections are coming back online as the DNS change propagates. That's usually fast, so the rest should follow very soon.
If a computer still shows as offline after a while, flushing its DNS cache makes it pick up the new address right away. The steps for Windows, macOS and Linux are
here.
Marius, thanks for the status page suggestion.
Thank you all for your patience.