Fair use hasn't been tested in US courts, at least not as far as backing up your own dvds, so the answer isn't perfectly clear cut. Most of us around hear interpret fair use to include the right to make copies for one's own use. The MPAA might disagree.
No one has been prosecuted or sued for individual backups of movies they own---not to my knowledge. Sharing such material, especially over the internet, is a different matter.
DVDFab Decrypter is a free ripper and is updated fairly often, so you might want to try it if you don't want to buy AnyDVD. If you need to reduce the size of the video once you've ripped it to the hard drive, you could use DVDShrink, which is also free.
DVDRebuilder is another option to reduce the size, though it takes much longer. There is a free version of that program as well.