Permissions denied to owner, although it is set to 666 in .ssh directory

Steeldriver is right. On directory you need also x access flag to be able to list files inside.

Fixing the directory using chmod 700 ~/.ssh should help you to get into this (correct) state:

$ ls -ld ~/.ssh
drwx------. 2 user user 4096 Aug 26 10:37 /home/user/.ssh

Also you should fix your keys using chmod 600 ~/.ssh/id_rsa and chmod 644 ~/.ssh/*.pub to get this:

$ ls -l ~/.ssh/
-rw-------. 1 user user  1766 Mar  7  2014 id_rsa
-rw-r--r--. 1 user user   415 Mar  7  2014 id_rsa.pub