Apparently there is some corruption in your root filesystem after all.
You can use the rescue mode without entering the "chroot /mnt/sysimage" command too, but then you will have a reduced set of commands available, and all pathnames of your files will have /mnt/sysimage prefixed to them. If that's enough to back up your files, then do it. Otherwise you'll need to find the problem that stops you from running the standard shell in chroot mode, and fix it.
I think "rpm --root /mnt/sysimage -Va" should allow you to verify all the files managed by the RPM package manager on your system. The command should list all the files that have been modified after they have been installed from RPMs. For some files (e.g. configuration files and log files), this is expected. But if you see something like /bin/bash or the system libraries in /lib have changed, you might want to reinstall the respective RPM.
It should be possible to reinstall RPMs in rescue mode without the chroot command, using commands like "rpm --root /mnt/sysimage --replacepkgs -ivh <pathname of the RPM>". Once the chroot command starts working, many things - including backing up your files - will be easier.
Alternatively, you can find some Live-CD-type Linux distribution, and boot the system from it.
(For example, Knoppix used to be good for rescue purposes: in the past, I've used it a few times for purposes like this.)
You may have to activate the VG and mount the filesystems manually, but otherwise the Live CD environment might be more user-friendly and have more tools available than the RHEL 4 rescue mode.