How to repair "sudo: /etc/sudoers is mode 0640, should be 0440?"

I think your best bet is to start the system in recovery mode (see here for instructions), drop to root shell and do chmod 0440 /etc/sudoers.

EDIT: as Mik suggested below, the following should work as well without rebooting the system:

pkexec chmod 0440 /etc/sudoers

Why might the command cuase this to happen? I suspect you ran the command while your current directory was /etc!

sudo refuses to work if the permissions of /etc/sudoers have been tamprered with to avoid compromising the system. You will have to fix the permissions using recovery mode unless you have another way of getting root access.