Chances usually aren't good. It all depends on what went wrong with the HDD. If it were just bad sectors where the master boot record is located, which would prevent it from booting up a computer, and the files are elsewhere on the drive, then maybe you could get them off. But if the controller failed, then you have little chance unless you got another HDD of the same vintage and swapped the boards with each other.
If the drive was spinning up and wasn't making weird scraping or ticking noises and hasn't been dropped on the floor or anything then the chances are pretty good that you can get something off of the drive. In my experience.
Everything sounded as it should, the only difference is it stuck at the white apple screen and then switched off. No funny noises or anything different other than just not booting up
Everything sounded as it should, the only difference is it stuck at the white apple screen and then switched off. No funny noises or anything different other than just not booting up
Well, it would still be worth it to get an enclosure or the necessary cable to connect it externally if you can and attempt to recover some or all of your files.
Then, if you buy an enclosure, you should also buy a new HDD and then use it as your Time Machine drive so that you won't have to lose anything again, as you could restore from a backup.