There are many different beeps and noises that a hard drive can make.
Some are downright weird.
Some of them are created by the motor trying (and failing) to get the drive spinning. Some are created by the actual drive head scratching across the surface of the drive, scraping your data away as it goes. Some are even related to both, with the head grinding away and preventing the motor from spinning. There are other situations where hard drives can make these noises but none of them are good noises for the drive to be making.
Given that your beeping started as your drive fell I have a feeling that the gyroscopic properties of the spinning platter combined the abnormal motion of the fall have caused your drive head to come in contact with the patter and that initial "beep" was actually the sound of the head grinding away part of the platter.
To give an idea of how easily this can happen there is an comparison I once heard between hard drives and airplanes (see page 15):
To give a sense of scale, the head would be like a Boeing 747 flying just 45 feet above
the ground at over 300,000 mph and counting every single blade of grass.
If the "ground" moves ever so slightly, as it would when gyroscopic motion tries to counteract the motor being twisted around quickly, then that plane is going to crash hard.
The continuing beep could either be the head continuing to grind at the drive (possibly stuck in a grove) or the motor being unable to spin the drive up due to friction.
You can try putting the drive in the freezer, but that's only really a (temporary) cure for a drive that works for a short while after powering up, then dies due to heat. It helps give you a little extra time to get data off but it's by no means a cure.