What tends to happen with very old hard drives is that the friction between the heads and the disk surfaces becomes too high for the motor to spin the disk. The term that is often used is 'stiction' as the heads stick to the disk surface.
Modern disk drives tend to park the heads on a small ramp at the outer edge of the drive platters. The heads are on spring arms that push the heads towards the disks' surfaces, but when the disk is spinning air movement very close to the disk surface keeps the heads flying very close to the surface without actually touching. The ramps lift the heads slightly away from the disk so there is no contact when the disk stops spinning. When the drive is shutting down, the heads are moved to the parking place, and are normally locked there so any shocks to the drive while it is shut down do not move the heads back over the surface.
Older disks used a different method. They had part of the disk surface that was designated as the parking zone. Often this was near the centre of the disk rather than the outer edge as in the ramp system. On these drives, although the actual data part of the disk is very smooth, the parking zone can have small pimples on the surface. As the drives spins down the heads make contact on the landing zone. The little dimples mean that the area of the heads that contact the disk surface is very small, and the friction is low so the motor can spin up. After many stop/start cycles, the dimples and the heads get polished and the area of contact and hence the friction increases. The problem first appears when you have to attempt a drive start up several times, but eventually the stiction becomes to high and the drive will not spin up.
Sometimes you can get a brief respite from the problem by taking out the drive and holding it so that the disk axis is in line with your arm and the largest two faces of the drive are at right angles to your arm. Next, rotate your hand back and forth quickly as though you are turning a screw driver. The sudden rotation of the drive casing will try and drag the disk around with it inside the case. The mass of the disks will resist this, and if this is sufficiently vigorous then it may overcome the stiction. If you are lucky, the next time you power up, the disk may spin up as before.
If you succeed, keep the drive running until you have copied off everything you need to recover. This might include cloning the drive onto a replacement.
As an extra bit of information. If the drive loses power suddenly while it is running, then energy in the spinning disks is captured by the drive motor acting as a generator and this power is used to park the heads safely before the disk slows down too much.
I hope this helps.