You did something really wrong: you gave everyone the power to modify the root directory and all of its subdirectories (there included /etc and /etc/sudoers*).
For this reasons sudo will refuse to run. From the friendly manual:
To help prevent the editing of unauthorized files, the following restrictions are enforced unless explicitly allowed by the security policy:
· Symbolic links may not be edited (version 1.8.15 and higher).
· Symbolic links along the path to be edited are not followed when the parent directory is writable by the invoking user unless that user is root (version 1.8.16 and higher).
· Files located in a directory that is writable by the invoking user may not be edited unless that user is root (version 1.8.16 and higher).
If you have root's password, please use su - to become root and fix back the permissions.
Otherwise you need to reboot the machine with init=/bin/sh, remount the / file system as rw (read-and-write) and, again, fix all those permissions. That can be tedious and error prone, indeed, but it's still possible.
As root you can start with:
find / -type d -maxdepth 1 -exec chmod -vc go-w {} \;
chmod -vc 440 /etc/sudoers