There are a couple of things that you could try. The first is to remove the drive from the system and then attach it to your current machine by means of an adapter or external enclosure, or if your current machine has the same connection type, you could install it in your machine. You would then have access to the data and could copy it off. You could also use another machine to access the data and copy it.
If the drive is a SATA drive, then there are devices which you can insert the raw drive in and it attaches by means of USB, it is called a Hard Drive Dock. Quite useful for accessing raw drives. There are also adapters for older drives, but external cases can also be used and don't always need to be put completely back together in order to use it, but cooling issues can occur, so fan can be useful to keep it cool.
Another method is to see if you can boot the machine. If so, then you could transfer the data to another device, such as a USB thumb drive, if the machine has USB.
Yet another method is to boot the machine using a LiveCD, such as Knoppix and see if you can copy the data that way.
The questions are whether the machine still works, but more important whether the drive still works.