So, since I didn’t find an appropriate answer while searching, I’ll write up the solution I found myself for others that may run into the same situation (and myself in a few years, probably):
Situation:
None of you cubes has DNS resolution anymore, but pings with known IP address work. The other computers in your home network are fine, it’s just something with your QubesOS. Starting up Firefox or something else in sys-net to investigate the issue doesn’t work. qvm-run -p sys-net -- ip a hangs indefinitely. Somewhere shortly a message about lack of disk space pops up. WTF?
Root cause (in my case):
My debian-11 templateVM, which my sys-net is based on, had completely ran out of disk space due to old kernels not being removed.
Solution (in my case):
Give debian-11 (or whatever templateVM is causing the issue for you) 1 GB more system space for the moment. Now starting up a shell in debian-11 works. df -h verified the issue. Switch to root via sudo -i then search for the wasted disk space via du -d 1 -h starting from / onwards, always cd’ing into the folder with the largest entry. Voila, /usr/lib/modules was the culprit. Run apt autoremove and go grab a coffee… Shutdown debian-11 and restart sys-net. Everything works again and they lived happily ever after.