Fixing var errors

Solution 1:

A short intro into Unix file permissions:

File and folder permissions are represented as 4 octets. The 1st octet is for special permissions. The 2nd octet represents what the owning user of the file/folder can do (this is almost always 7). The 3rd octet represents what the owning group can do (usually 5). The 4th octet represents what everyone else can do.

Each permission is represented by numbers. 4 is for reading, 2 is for writing, and 1 is for running/opening (running a script or opening a folder).

The first command that you ran couldn't have been chmod -R /var, because it doesn't specify the permission to set. If the permission that you chmod'd is indeed 0700, then that means only you have full (read and write) access to the folder; the system can't even peek inside the folder. I'm not sure how the first instance of apt-get worked, though.

Fixing the permissions should cause your system to fully boot (and likely restore apt-get). The solution I can think of (assuming you can't get to a tty) is to run a live session from the live CD and, on /var of your hard drive (not the live session), run:

chmod -R 0755 /var
chmod 0377 /var/crash
chmod 0755 /var/log
chmod 0777 /var/run
chmod 0777 /var/lock
chmod 2775 /var/local
chmod 3777 /var/mail
chmod 3777 /var/metrics
chmod 1777 /var/tmp

This should get you back on the right path.

Also, I strongly recommend you don't change the permissions of system folders (basically, anything outside of /home).