If it's a drive in an enclosure, I'd start with seeing if you can remove it, and plug it directly into a computer or transplant it into a different external enclosure. If it's USB powered, often the USB port will not provide enough power to run it, and you'll get funny noises/no response from the drive.
I've had a pretty good success rate with freezing drives to recover data from a dying/dead drive. Much simpler and cheaper then most other methods of data recovery.
Freezing the drive shrinks everything inside, sometimes freeing up stuck/seized/trapped components, or keeping an overheating component from shutting the drive down right away.
Stick the drive in a ziplock bag(to seal out moisture) in the freezer.
While it's freezing, set everything up to quickly transfer your files off the drive if it does work. After a couple hours in the freezer, plug the drive in and see what it does. It may only work for a few minutes, for hours, or weeks. Pull your data off as fast as you can, if it stops again, re-freeze it and start where you left off. I've frozen drives 3-4 times to get everything off.