All moving air does is increase the rate of the heat transfer from the higher temp substance to the lower temp one. If the air outside is hotter than your body temperature then a fan will be heating you UP not cooling you DOWN.
Without electricity or chemical reactions the only way to stay cool in high heats is to use evaporative effects. Evaporation is endothermic (takes in heat) and actually cools down the thing being evaporated off of. So you can basically soak your shirt and as it evaporates it cools you down. There are things called 'evaporative vests' which can cool you off for hours.