As with any electrical installation, the wire size used depends on the expected maximum current the wire will carry. That cannot be determined by voltage alone.
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The size of conductor needed will be a #14. It should be copper and have an insulation factor of 90 degrees C. The fact that it is a three phase load does not enter into the calculation of the wire size.
No a 208 volt outlet does not need a neutral. 208 volts is the line voltage between any two legs of a three phase 208 volt system.
A 500 MCM copper conductor will limit the voltage drop to 3% or less when supplying 350 amps for 150 feet on a 208 volt system.
what size wire would be required for an ak012 with a power supply of 208/240-3-60 hz
The wire size depends on how much current it will conduct.