By your description this is not a well, it's a catch basin (like a cistern). It is only to catch water. Many years ago spring, stream and dug wells were reasonably safe but with todays population and chemicals many of these same sources of water are questionable.
I'd pour a gallon of household bleach in the catch basin and stir it up a little, turn on the pump and pump water to each faucet, when you smell chlotine (Bleach) turn off the faucet and go to the next including the showers and tubs. Then let it set for 24 to 48 hours then pump all outlets until you no longer smell the chlorine/bleach. This should purify everything. However it won't purify any incoming contaminated water. Be aware that in todays society not every unsafe bacteria can be removed by chlorine/bleach.