There's nothing you can download or update that will change the drive. The only two things you can do is to either open up the eMac and replace the drive with a DVD-RW Superdrive, or buy an external DVD burner with preferably a Firewire 400 connector and USB 2.0.
Depending on what kind of files they are, you can try zipping them before trying to get them onto a CD. Some files don't compress too well but if you select a file that's on your desktop, and either right-click on it, or hold down the Ctrl key and click, you'll get a menu that says 'Compress xxx' or maybe Archive. Depending on the OS you're using, the same command is in the File Menu under Archive or Compress.
Give that a try.