I created my own world a while ago and now I'm busy populating it. One problem I just can't fix is that one of the households will become a pet adoption center or something (in German it's called Nachbarschaftstieradoption), and the household that currently has this label is not playable anymore. How do I prevent this? And once it has happened, how do I make my household playable again?

If not averse to using NRaas mods, prevention is easy. Overwatch has a setting that stops the Pet Adoption Centers from spawning.
NRaas > OW > Settings > Stop Pet Adoption > True
(this doesn't stop all pet adoptions like its name might imply, only the often broken and much hated Neighborhood Pet Adoption Center scenarios)

If using NRaas StoryProgression, then there are no Neighborhood Pet Adoption Centers. The mod blocks them for all users, no questions asked.

Once you already have one, you could try using that setting on OW to see if that shuts it down or you might have to wait several sim days for it to time out. If the scenario is stuck, then sometimes there isn't much that can be done other than TAing (force deleting) the entire household by way of MasterController and bulldozing the lot. Perhaps others here have found a better way to cancel it. From where you are now though, do you have a way to revert to a prior save from before the scenario started and replay from there without losing too much progress?

Thanks, I've had Overwatch for a while now but never knew that's what this option did! It removed the Pet Adoption Center and I can now play with my household normally again