The hard drive in your old machine is probably a standard ATA/IDE drive (what they call PATA now, since SATA came along). You probably won’t be able to boot a new machine from the old hard drive, but you can still access the data on it. The easiest way to make the old drive accessible on a new machine is to put it into an external IDE-to-USB enclosure. NewEgg has tons of these, and I’ve seen some for under $20. Once you have the drive in the enclosure, just plug it into the USB port on another machine and it should mount up as another hard drive.
Keep in mind that any viruses on the old drive may contaminate the new one, so be sure you’re running antivirus software.
If the old drive is a Windows 98 drive, your documents will be living in a folder called “My Documents” at the root level of the old drive (for example, E:\My Documents). On anything Windows 2000 or newer, look in E:\Documents and Settings\[your old user name]\My Documents. You can copy what you need over to the new “My Documents” folder whenever you like.