Why did the Greeks run naked?
Back in the days of Socrates and Plato they did. Wrestling was
always done in the "buff". In fact all the sports performed in the
Olympics (Which originated in Greece) were performed in the
nude.
During the Archaic period Greek soldiers wore pretty much what
they wanted. However, during the Classical period the state
organized and required- to a degree - a standard of battle dress,
which included helmet, chest protection, shield, sword, etc. I am
not aware of any Greek worriers who fought naked. All vital parts
of the body were considered for protection to include private parts
of the anatomy.
There is a reference made to Aias, son of Oileus in the Iliad,
whose armies fought without armour, though this does not mean they
fought without clothes.