PostgreSQL pg_ctlcluster won't work
Issue solved.
Unknown to me, one member of our team deleted the main log file of the database, because it had grown quite a bit. Apparently it was deleted completely instead of being truncated. Then a new log file was created, but for some reason with the root as owner.
Simply changing the owner back to postgres like so

chown postgres:postgres /var/log/postgresql/postgresql-9.4-main.log


solved the issue.