Linux has many ways to make a personal DVD backup if you are comfortable using command line tools. If however you would rather use a graphical interface, there are three main contenders - K9Copy, AcidRip, and DVD::RIP.

In order to read commercial DVDs, you must have the libdvdcss2 file to read the encrypted files. This file is not in the default repositories, instead you will have to find the repositories that hold the media packages that cannot be included in the installation packages for legal reasons. For Ubuntu that is the Medibuntu repository. For Fedora, you can find the library in the DAG repository.