There are several ways you can do this, assuming the HDD is still functioning. Probably the easiest is to find an enclosure that fits your drive type, thereby turning the internal HDD into an external USB one.
You can certainly take the hard drive out. You can use an external enclosure to connect your hard drive to any other computer using a USB cable. Just do a search for external enclosure. You'll need to know if it's IDE or SATA, but you'll know that once you have it out.
Actually, you don't need a whole enclosure if you just want to hook it up and pull the old data out. You can get an adaptor that goes usb to both sata and ide:
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that looks like just the thing, thanks
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