Restarting Steam made the correct times appear again.
I think restarting is just a temporary fix. Mine was at 220 hours because I leave my computer open at all time. Resetting Steam does indeed put the time back to normal, but now it's going up again.
There's a problem here but this is not a Linux issue, it's a general Steam issue and will be tracked in the regular forums.
Nothing extracted yet.
It looks like there's a bug in the play time tracking server. Even the games that I never really played (just tried them out for 2 min) now show that I played them for ~60 minutes.