In short, I bought a new external hard drive (A Seagate STDT4000100, part number 1KBAA3-570. A 4 TB external.) and used AceBackup 3 to put a lot of data on it.
Things went fine for a few days, until I woke up after a nap and found the hard drive was making that sound (my computer was off at the time). I unplugged it and after a while I plugged it back in and the sound went away.
Later, the sound came back and I tried unplugging it and then plugging it back in immediately- the sound came back.
Unplugging it and then plugging it back in after a while made the sound go away.
Seatools tests seem to be okay so far (I am currently doing the long generic test, but the HD passed the other 2 tests).
Any ideas? Should I be worried?