On my site automatic backups are made with UpdraftPlus.
Actually only 3 backups are always kept, older backups should automatically be deleted.
In this case old backups were not deleted automatically, for whatever reason.
It seems that the webspace quota was reached.
After I deleted some backups manually, the site worked again. The error message was gone.
… and then, when I submit a search, I get the same error message again on https://www.greenfilmnet.org/members/, and there is just empty content (no search result).