If you get the message:
sudo: can't open /etc/sudoers: Permission denied
Then check the following:
- Are the sudo permissions correct: /usr/bin/sudo should have 4111 (---s--x--x root root)
- Are the sudoers permissions correct: /etc/sudoers should have 0440 (-r--r----- root root)
- And the not-so-well-documented: The directory containing sudoers (usually /etc) must be readable by the group that owns this directory (not just the user!!)
ensures that the file permissions are correct, but it does not check the /etc-permissions.