psql: Could not connnect to the server: No such file or directory
Alright, I found what was wrong with all of this.

First, by running /usr/bin/pg_lsclusters I could see that the cluster was indeed not online. Then I checked the /var/log/postgresql/postgresql-9.5-main.log file to look up what was the specific error that had caused the cluster to not be brought up and there I could see that my ph_hba.conf file was, somehow, impossible to read.

I don't know how or when the permissions changed but they did. I chmod 644them and now everything is awesome.