I just stole the jet. Super easy to do.
My suggestion is to make sure you have a decently fast car to do this. Use Trevor for this for the best odds.
Haul ass breaking through the military base front gate, make your first RIGHT, then your first LEFT. Take it straight down to the hangar. There is a jet inside the hangar and the hangar will provide better coverage from the tanks. Jump in the jet, full throttle take off and head to the airport hangar you bought with Trevor. Park it inside the hangar and then go get yourself killed. Then the jet will be saved in your hangar.
Now, the decision I have to make it do I take the Jet Fighter or my Learjet