Try booting with bootflags=ro,recovery in grub (with the latest
possible kernel), or mounting with -o recovery from the livecd
(likewise).  If it works, then you're done, you should be able to boot
normally after a clean umount and shutdown.  If it doesn't, post dmesg
from the attempt.

I'v been told this is missing relevant details.
The original kernel version was 3.2.0-something (standard Ubuntu 12.04 LTS).
I've since upgraded to 3.7 but this has made no difference.
Right now I don't have the dmesg, I'll post it later.

Currently I've been able to mount the partition with btrfs-restore and am
trying to rsync it on another ext4 volume.

Terminology note: btrfs-restore doesn't "mount" anything, it just
copies files directly from a device.