So you have a supply of water at 50°F - that’s 10°C (an optimistic temperature in many places). And you are using it to try to maintain a body temperature around 37°C. Let’s assume you do so by transferring heat from your body into the water enough to raise its temperature by 25°C - that’s probably overly optimistic given that it will also be warmed by the environment and the rate of transfer will be reduced as the temperature increases, but it’s a working number.
The specific heat capacity of water is about 4kJ/kg/°C so a liter of that cold water can absorb 100kJ of energy in the process of warming up. Once it’s warmed up it can’t do any more cooling.
Your body, when not working hard, produces about 200W of heat. That’s 200J per second.
So that liter of water can absorb your body’s heat output from about 500 seconds of being alive. That’s a little over eight minutes.
So that means the cooling needs for one human for one day can be met by about 170 liters of cold, 50°F water. That’s about 44 US Gallons.
If, however, that water can evaporate - well, the latent heat capacity of vaporization of water is over 2 thousand kJ per kg.
That’s why the wet bulb temperature matters. If water can’t evaporate, that cold water can’t absorb body heat except by just warming up.