Cannot start psql with user postgres “Could not change directory to /home/user ”
I solved the problem with the help of a relative.

The first problem was that my /home and /home/user folder didn't have the right permissions.

We set them correctly with chmod og+rX /home /home/user.

Then I forgot, that I created a new PGDATA directory in /home/user. And this directory had the wrong permissions. So we set them again for the user postgres with chown postgres:postrgres -R /home/user/PGDATA.

I decided to place the PGDATA directory in /home/user because of huge log-files that where produced during raster data queries with my PostGIS/PostgreSQL database. Sometimes I'm using Sublime Text 2 for queries and I quit them without stopping the database process and then the logfile became larger and larger and I was running out of space and nothing worked anymore.