Couldn't you just eat ice to keep cool? Heat from blood transfers to the ice as it melts and warms in the stomach, then it leaves the body completely through urination.
Indeed, we even still colloquially quote/measure the cooling capacity of air conditioning in "tons".
A ton of cooling being defined as 12000 BTU/hr, or the rate implied by the latent heat of fusion of 1 ton of ice at 0°C being consumed via melting over 24 hours.
Similarly you could put a giant block of ice in front of a fan and sit there; most people would call this a from of AC though.