How to chmod without /usr/bin/chmod

Boot another clean OS, mount the file system and fix permissions.

As your broken file system lives in a VM, you should have your host system available and working. Mount your broken file system there and fix it.

In case of QEMU/KVM you can for example mount the file system using nbd.

The good news is that all your data is still there. The mixed news is that your system installation may or may not be recoverable — it depends where chmod stopped.

You will need to boot into a rescue system to repair it. From the rescue system, mount your broken installation somewhere, say /mnt. Issue the following commands:

chmod 755 /mnt
find /mnt -type d -perm 644 >/mnt/bad-permissions
find /mnt -type d -exec chmod 755 {} +

The first find command saves a record of directories with bad permissions into a file. The purpose is to see where permissions have been modified. The second find command changes all directories to be publicly accessible.

You now have a system where all directories listed in /mnt/bad-permissions and all files in these directories are world-readable. Furthermore files in these directories are not executable. Depending on which files were affected, this may be easily repairable or not. See Wrongly set chmod / 777. Problems? for what you can try to get the system functional, to which you should add

chmod a+x /bin/* /sbin/* /usr/bin/* /usr/sbin/* /lib*/ld-*

But even if you manage to get something working, there's a high risk that some permissions are still wrong, so I recommend reinstalling a new system, then restoring your data. How do I replicate installed package selections from one Debian system to another? (Debian Wheezy) should help.