Does your computer have a CD-ROM and do you have access to a CD burner (either on that computer or a different one)? If so, you can use a linux live cd to access the file, then copy them to another device. You just put burn an image to a CD, pop it in the drive, then depending on your boot settings it should start up automatically. You can then access the hard drive and copy the files wherever you want. The livecd itself doesn't actually install anything on your hard drive, it a standalone operating system that runs entirely from the CD. It doesn't leave anything behind when it is done besides the changes you explicitly make.
I recommend
Parted Magic. It is free, very small, light on resource usage, works on most computers, and has a good set of hard disk tools. It is also compatible with both windows NTFS (what windows nt, 2000, xp, and later use) and fat/fat32 partitions (which windows 98, ME and earlier use).
So in this case you can connect the drive, put the disc into the computer you want to move the files to, then run the livecd. It will give you some options when booting up, just go with the defaults (just hit "enter" when you see a prompt). Once it boots, you will be in a graphical desktop not fundamentally dissimilar from windows. there will be icons on the desktop for the various partitions it can see. Double-click the drive you want to move the files from and the drive you want to move the file to, then just drag and drop. I've done it even from drives windows cannot access.
If that doesn't work, then the hard drive likely is corrupted. There are tools on the disc to recover damaged partitions, but lets not get into that unless you absolutely need it.
Depending on how much of the hard drive is used, you can even copy the files over to a new partition without deleting them. This involves resizing the partition, creating a new one in the empty space, moving the files over, deleting the old partition, then resizing the new partition to fill the space. This is not guaranteed to be 100% safe but I have done it dozens of times and never had it fail on me before. It is better to just get the files off, though.