The first step is to download DVDShrink. It's pretty easy to find and you can download it here or here. After that, double-click the installer; the default options should be fine for most users.
After you've installed DVDShrink, open the program. You should get a window that looks something like this:
Next, click the "Open Disc" button at the top and select your DVD drive from the menu (usually the D: drive in a standard Windows environment):
DVDShrink will now begin analyzing your DVD. This usually takes around 2 minutes. You can uncheck the box labeled "Enable video preview" if you are worried about spoilers:
When it's finished analyzing, you should end up with the DVD structure in the left panel and some compression options in the panel on the right, like this:
Next just click on the "Backup!" button at the top:
You should get a menu like this:
From the "Select backup target" pull-down menu, select "Create ISO image file" and then click the "OK" button:
This will make an exact copy of your DVD, shrunken down to fit a single-layer DVD. There are many other options you can fiddle with in DVDShrink, such as removing sound from menus or reducing the quality of unimportant features to improve the quality of the main movie, but they're not really *necessary* so I'm not going to cover them here.
When DVDShrink finishes encoding, you should have an image file (ends in .iso) that will fit on a standard 4.7GB single-layer DVD. To burn it, you need to use an external program with image-burning capabilities, such as Nero or Alcohol 120%. I personally use the shareware demo version of a program called DVDSanta available free from various places, including here.
If you get this error:
then your DVD uses an unusual copy-protection mechanism that DVDShrink cannot bypass. If this happens, you'll have to rip the DVD using another program first, such as DVD Decrypter, and then use DVDShrink to resize it (Note: this error may also appear when running the program through WINE. Thus, Linux users are encouraged to use DVD Decrypter first).
Copyright notice: using this method to rip commercial DVDs, even if you own them, may be prohibited in your country. It is best to research the laws in your locale before following these steps.