BASICS OF RESTORING AN OLD REK-O-KUT TURNTABLE

Check to see how it runs.  If it runs, start cleaning it up and re-lubricating it.   Cleaning is pretty simple, but a little messy. You may or may not want to get involved and pay others, but it is basics:
 
Ronsonal is a great solvent that can be used with paper towels to remove 40 years of surface dirt and oily film from all cosmetic surfaces.
 
Most oil has turned to something more like grease, and most grease has turned to something more like glue.  Again paper towels and Ronsonal can be used to clean areas where you see lubrication applied. Then use just a drop or two of oil (common motor oil - Mobil 1?) to appropriate sliding surfaces, or a little grease (like Lukas White Lithium Grease or Ultralube) to where grease had been used. Add a drop or two to the motor oil inlets if you can find them.  Always be careful to use a minimum of oil and not drown it.
 
Parts that may be required are:

A motor start capacitor.  If the motor won't run.  That could be the cause.
 
Motor mounts.  If all works fine and you don't hear obnoxious motor noise coming thru your speakers, you probably don't need them right away.  Then again....

RIM DRIVE PLATTER:
 
Rubber idler wheel. If it is hard, you may hear obnoxious motor noise coming thru your speakers.  Try our links or google for someone who can rebuild yours or sell you a new one.

Hint:  If you can hear the motor noise thru you speakers, replace the motor mounts first.  If no cure there, do something about the idler wheel.

BELT DRIVE PLATTER:

The original cloth/rubber-impregnated belt is junk.  Get a new rubber belt.