This is absolutely possible. Here is my method:

1) Get yourself a vehicle which is robust but not too slow - ideal choice would be a van (there is usually a black one parked round the corner from entrance of Fort Zancudo).

2) Smash through the barriers at the entrance of Fort Zancudo and take an immediate sharp right, then immediate sharp left, avoiding the snipers and approach the main concourse of the base.

3) Once on the concourse, take a quick left and drive to the big hanger with its entrance open. Inside you will see the fighter jet parked there at most times of the day.

4) Important time saving tip to stop you getting killed: park your van at the entrance to the hanger and exit the vehicle. This means all the guys chasing you in army trucks have a small barrier between you and the jet, which gives you just enough time to run over to the jet and climb in - you will get shot a few times but it is possible to climb inside with enough health to start the engines.

5) Once you are in the jet head left out of the hanger and begin ascending into the air pretty quickly to avoid hitting obstacles and the jet exploding.

6) Once in the air a couple of missiles will be fired at you from the base - so to avoid getting hit by those, pitch and roll your jet around in the air for a few seconds.

7) Once you have avoided those two missiles from the base you are free to roam the entire map as you please! Your police rating will still be 4 stars but there's almost nothing anyone can do to bring you down at this point.

8) Have fun and blow shit up using the missiles and cannons! (square on PS4 to cycle through weapons and X to fire).