When you say it will not reset do you mean you hear the pop of it turning itself off as if there were an actual ground fault?
For testing the circuit....
You should have zero voltage between ground and neutral and full voltage between hot and neutral. Your hot I believe is directly shorting to neutral...therefore it is shutting itself off. You have a direct short I believe or a faulty GFCI breaker. Go to the hardware
store and get one of those 120 volt outlet testers. The light pattern on the tester will tell you what is happening. Remove the GFCI outlets from the circuit to confirm whether or not they are the problem or the circuit itself is the problem. You will obviously have to wire in a standard, non GFCI outlet in order to use the 120 volt tester. Make sure the circuit is on its own breaker at the source for safetys sake. Also, remove any loads from the circuit.
Senior mechanino is probably right...salt or something somewhere is causing a partial short between hot and neutral or hot and ground. Salt
acts as high ohm resistor when it has crystalized and is still a little wet.
Also, dont use wire nuts...use crimp connectors. You probably already know that though.