There are places to go to recover the data on the drive, but it's a very expensive process and only worth it to do if you have lots of cash, and the DATA is THAT critical to you. Many businesses carry insurance that pays for such data recovery, but even the insurance costs a lot.
I have taken external drives apart before, pulled out the drive, and hooked them up internally to a computer with success.
If the logic board on the HDD is bad, you can often buy a matching drive and swap the boards between the drives to recover your data. This only costs the price of an identical drive, and costs far less than professional data recovery does.
If, as you say, the external drive was getting hot, it's cooling fan probably failed. That excessive heat probably ruined the power circuitry inside the box. The drive inside is probably still good.