Ran into some issues ripping backups of DVD's on Ubuntu. First one worked fine but the second and third failed. I did a little research and quickly found the solution, so figured this might be a good time to document everything I know about the subject for future reference.

First of all, certain libraries are not included in any linux distro due to legal restrictions. Fortunately in Ubuntu, getting those libraries is pretty simple. This page has good instructions on how to get them. How the first DVD I tried worked without all this is beyond me, but I'm not too worried about it.

Next, install k9copy, xine, and k3b via System/Administration/Synaptic. k9copy is the Linux equivalent to DVDShrink.

Now, insert the DVD and use k9copy to backup the DVD to an iso file on your hard drive. Be sure to select the 'Keep Original Menus' option.

To watch the DVD directly from the iso, with the physical DVD safely stored on the shelf, run...
xine "dvd://full/path/from/root/to/file.iso"

To move the iso onto another physical DVD, use k3b.

Note that k9copy and k3b are technically KDE apps, but they both run just fine under gnome.