Air in Water Lines - Jet Pump System
There could be 2 problems but I would check your pressure tank first, it sounds like your bladder or diaphragm has failed, if a really old system a diaphragm tank needs to be replaced, you can check the air pressure at the tank there is normally a shrader valve  that you can check the pressure with a car or bike tire pressure gauge, if the diaphragm has failed if you turn the pump off and drain the system by opening a faucet and you get a bunch of air just prior to nothing more time to check the tank pressure it will probably close to 0 psi but if it has ~40 psi (or a couple psi less than the turn on value of the pressure switch the tank is good and you have not purged the air out of the system, air in the pump side creates air in the jet and more air in the tank, it can be tough to get the air out but high flowing a couple of faucets usually will do it. If the tank pressure goes to zero or close you can temporarily get the system running by putting some water in the tank, charging the tank with air a bike pump or air compressor is used to fill the tank to just below the turn on pressure for the pump, many times 40 psi. Now let the pump fill the tank, I have had this work for a short while so I could replace the tank on the weekend, if left in for a while you might find your water starting to taste bad because of exposure to air in the tank. So check your pressure tank, it could be a cracked line in the well or at the jet but if you are reaching pressure with the pump shutting down and pressure is solid it is probably the diaphragm or bladder in the tank that has failed. Note in most cases it is cheaper to replace the tank but some bladder based tanks the bladder can be replaced for less, but it is a fair amount of work and a new tank has a better warranty.