HandBrake is an open-source, GPL-licensed, multiplatform, multithreaded video transcoder, available for MacOS X, Linux and Windows.
It is my tool of choice when it comes to backing up my DVDs. It allows me to shrink DVDs to a relatively small size and convert them to a web-playable format, such as mp4.
I use the Command Line version of HandBrake simply because it is more efficient and I love typing geeky commands in my terminal window. So here it is, the simplest command to convert your DVD into an .mp4 file is this:
HandBrakeCLI -i VIDEO_TS -o myVideo.mp4 -e x264 -b2000 -B 192
That will encode a source video located at the path VIDEO_TS to an output file called myVideo.mp4. It will use x264 with a bitrate of 2000 to encode the video, and encode the audio as 128kb/s AAC.