I make copies of DVDs that we buy for my girlfriend's 8yo daughter. She has a tendancy to not take care of them, and considering she's lost two copies of Finding Nemo, I think making backups is a good idea. On the Mac, you can use
Mac The Ripper to rip the movie to your drive. Then, you'll have to use something like Popcorn (which is not free, btw) to make it fit on a 4.7gb DVD.
On the PC side, I find things to be a bit easier and even better. Use
DVDshrink to rip and compress the DVD. If you have Nero, DVDshrink will integrate with it and automatically burn your blank. Nero comes bundled with a lot of DVD burners, so there's a good chance you have it already. I find DVDshrink to be better because you can uncheck things you don't need to keep on the copy (such as languages, special features, and trailers) which leaves more room for the main movie and results in better picture quality.
Newegg sells Popcorn for $42, and Mac The Ripper is free, but if you already have Nero on your pc, I think that is the way to go.