My idea on this is to use well water to cool the frig coils. Like a heat exchanger so the well water stays safe. That should improve the effieciency of a normal frig compare to cooling the coils with ambient air.
I would say if you just wanted to use the well water itself. Measure the temp to see how cold it really is. If cold enough, take a chest type freezer, insulate very well on the outside. Place a safe small radiator inside and a small cheap low power fan, maybe 12v computer types.
Run cold water through radiator and circulate the air inside the chest through it with the fan.
You should be able to get it as cold as your water supplie. How long it takes would depend on many factors, size of chest, radiator, volume of water etc..
Beware some radiators may still use lead solders, it may be considered illegal to put your water back into the well etc...
If you use a coil of copper tubbing and use a sealed loop system you should be ok and be able to use a low powered pump.
You could use an normal upright type fridge, but you lose so much cold air everytime you open the door I dought it would keep up very well using just well water for cooling, or you would need such a large cooling system, pipes, pump, ect.. it's probably not worthwhile.
You could try the old Ice Box methode of cooling and just put the cold coils, radiator, in the top of the box and let convection work. Cold coils cool the air at the top, it sinks, warmer air rises cools sinks. No power needed other than the pump for moving the cold water. But since your working with mabe 40-50 degree water, not ice, I don't think it will work very well.
My total opion on this really though.
This would be a good Idea for cold drinks, vegatables, and other things that keep well like eggs and cheese maybe. I don't think it will be cold enough to use as a normal frig for long term (normal) storage of meats and such that spoils fast. Then use a smaller more efficient normal type frig that gets real cold for things like meats and milk that spoil fast. Then you won't have to open the power hog very often and it should seldom run, plus you can use a smaller one. Use the water cooler frig for all the constant usage items like soda's beer, lunch meats, catchup, and things that get used fast often or keep well.
Like a dozen eggs should last a week or more I would think, so keep about 3 days worth in the water frig and the rest in the power hog.