Depending on where you are, (in temperate areas), your ground water is somewhere between 45 and 55 degrees (F). Water takes a lot of energy to heat up, so, yes you could use your well water to cool your house, depending on the size of the house, how tight it is, and how much water you can pump up.
First you have to get the water up from the bottom of the well (solar or wind, I'm guessing). Once you do that, if you just run it in a long looping route in the house and it should suck up heat. It would probably cool more effectively if you mount it up higher than the floor. Heat rises, cold sinks. If you cool the floor, your feet may well be cold before your head stops sweating. Cool at ceiling height and you cool the whole room.
The main things to consider are surface area and air flow past the cold line. The greater the surface area, the better the cold can absorb heat. The greater the air flow, also, the beter the cold can suck up heat. (think of a car radiator, lots of air flow, lots of surface area, except with a car, it's doing the opposite, the water is giving heat up to the atmosphere).
You could get some scrap radiators, stacked one in front of the other, with a fan moving air through them. (using what you can find, Automobile, Old drained A/C units, fridge radiators). The cold water should feed into the radiator furthest from the fan, then feed the others. Another alternative might be mounting them a few inches below the ceiling, parallel with the ceiling. That way as air cools, it will drop down and make a natural air convection.
POSSIBLE PROBLEM WITH THIS SUGGESTION: People used to poisoned when unscrupulous bootleggers in the 30's used car radiators to make whiskey, but I'm pretty sure that was because the alcohol and maybe some small percentages of acid leached lead out of the radiators and put it in the whiskey. I'm 85% sure water would be ok, once the radiator is well flushed out. Might not hurt to double check on that, if you're going to drink it)
Now that I think of it, it may be easier to break down and buying as many finned hot water radiators made for a house as you think you'll need. (the kind that are behind a baseboard on the wall). Only, since you want them to cool instead of heat, I would mount them up high, and maybe a few inches from the wall (condensation on the wall might be a problem (otherwise). My WAG (wild assed guess) is that you might want a little more to cool than you would use to heat a house of similar size, if you were using hot water heating).
If you are pumping fresh well water through the system, a radiator may eventually clog up with mineral deposits. (warming and cooling water changes it's carrying capacity for different substances, and I'm not sure off the top of my head if heating it up would increase or decrease it's capacity. If it decreases it, it may lead to mineralization. You might check on that.).
Where are you at? If you're in a humid area, cooling the air might lead to mildew and mold issues. You will at least get water dripping off of you radiators, which is simply a problem to be solved. In a very dry area, as was mentioned above, some kind of swamp cooling system might works well. (If you have a hot, dry breeze blowing through a window, pumping water up and letting it trickle through a loose mat or something the breeze blows through in front of a window will give you a very creditable swamp cooler imitation).