No, you don't need another Compaq computer.
While you could remove your harddrive from your computer and install it internally into another computer, I would suggest that you buy an external, USB enclosure for your harddrive. Just make sure that the enclosure is the right size (for 3.5" drive) and has the right interface for your harddrive (IDE or SATA.) If you do this, you can use any computer with a USB port to copy your data. Cost = ≈$20 - $40.
Good luck.