But if you can hand throw an ICBM, any distance at all, you win.
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No mechanical augmentation? Because using an atlatl would still be a form of hand-thrown, mechanically extending the throwing lever (arm length) but still muscle-powered and hand-held.
Longest atlatl-assisted spear throw was 850 feet. Decide for yourself if you want it to qualify as “hand-thrown” per the OP.
I was going to say, imagine using an atlatl to assist throwing that Aerobie Pro thing…That would be crazy.
If you go by “muscle-powered and hand-held,” bow and arrow would qualify. I don’t know the record for that, but found some unreferenced comments online saying it is “1,222 metres by Don Brown in 1987.”
The classic sling beats that with a 1434-foot throw.
Going a bit OT for humor’s sake: This guy was able to throw a discus farther than Hercules!
I thought those were called sticks 
A buddy once threw an Aerobie off Half Dome in Yosemite - almost 1500 meters to the valley floor…
Michael Collins dropped a Hasselblad Super Wide C during a Gemini mission that stayed in orbit about 3 years. I think that’s the record.
Actually the baseball also has the advantage of lift. It’s going to spin to travel any substantial distance. A non-spinning knuckle ball would not travel anywhere near 400 feet.
Very rare thing: (baseball) Line drive to center field with no spin. 100mph+ knuckleball with room to dance.
Or one of those jai-alai throwers.