I believe that this phrase is used when one is under stress or in a stressful situation. For example if your boss wants you to finish a report in about an hour you could tell your friends "I can't talk right now. I am under the gun to finish this report."

Under the gun is a phrase from track & field races. When starting a race, the runners take their places on the track in their assigned lanes. They have a few moments to check their shoes, stretch, etc. The starter usually calls them to attention to be sure they are ready to race. The runners take their first positions - they crouch & place their feet in the strting blocks. The starter raises his gun. At this point the runners are "under the gun" & must start the race when the gun is fired. After they are put under the gun they may not ask for a time out to fix anything. The shot fired by the starter is the deadline that is fixed & immovable.

"I am very much under the gun" = "I feel a lot of pressure to get this done in time."

Wow! Jura, I never realized that was the origin of that phrase, but you're right. And in common use it means what the others said.