Rather than RAM (default) or the root partition (/var/ -- far too small!), I used a high-capacity USB drive. Make sure this USB drive has enough free space before starting. You need at least X GB of space if your volume is 8TB like mine was. So, I'm guessing that this is why the fsck fails in the first place - at least, this is what my logs indicated when it said "Error storing directory block information (inode=90885846, block=0, num=6983948): Memory allocation failed"
Get root access to your NAS. I had already enabled root ssh before things went haywire because after the volume failed, FrontView would not load.
Attach a high-capacity storage device to the NAS via USB. Wait while the ReadyNAS loads the drive.
Download and compile the latest e2fsckprogs from the internet:
wget cd ./configure make