That's how a fridge is created for a scouts camp:
Dig a hole in a shady location outside and put the stuff to refrigerate into it. After that you will pin a tarp over the hole and keep it wet by pouring a little water over it. Through the continuous evaporation, it gets actually really cold inside the hole, even during summer. Out of my own experience I can say that it is definitively effective enough to cool down drinks, especially for a medieval setting. Instead of a tarp, you can just take a large skin. This might be even more effective as it has a bigger surface.