How does one go about backing up older CD-ROM games from the 90's?

Solution 1:

CD Burner XP (which also works on Windows Vista and Windows 7) can create a .ISO file for you (or just use its built-in CD Copy feature) using the "Copy or grab disc" option on the main screen.

  CD Burner XP (free software)
  http://www.cdburnerxp.se/

After that, you can use that .ISO file to burn a new CD.

I typically do this for my kids to have a copy to handle, that way if they wreck it (it's always an accident because they're just kids, I swear!) I can just burn a new one for them.

Solution 2:

In my opinion, the easiest way would be to save the CDs as ISOs; this way you'd have a perfect copy of the CDs and you'd be able to save them anywhere you'd like. Given the fact that the games you want to save aren't that new, the size of the ISOs would be relatively small, so storage would not be an issue.

Here's an article explaining how you can save a CD as an ISO image (there are probably other ways to do it).

Solution 3:

It's worth noting that if the CDs have some sort of copy protection, ISOs might not work. A more "raw" copy of the CD will be required.

I bought a ripper / burner from Alcohol Soft years ago which serves my purposes well (but it's not free).