37 is over the age for commisionning except the Navy needs nurses so they routinely grant exemptions for a lot for medical professionals. If they take you, you get to go to Oyster School in Newport - that's Officer Indoctrination School where they actually teach you how to salute and what the different Navy ranks and ratings mean. It's for doctors, lawyers, nurses, and other genteel staff professions. After you complete this, you get a commission as a Lieutenant in the Nurse Corps.
I went to Officer Candidate School at the same base, and in the middle of winter, I had guard duty in the middle of the night. On my rounds, I checked the gym - the Oysters were playing volleyball. A couple of really gorgeous nurses or lawyerettes wanted to know if I was going to stick around for the water balloon fight and ice cream after the volleyball game.
I looked at them, looked around at everyone nice and warm and happy playing in the gym, sighed, reshouldered my Springfield drill piece, and went back into the snow.
If you do well, in about 12 years or so, you'll be a Commander probably pulling down $100K with pay, allowances, incentives and nursing bonus.
The story that I heard about NAS Sigonella is that everyone who goes there for duty takes a rock with them upon departure. This way, one day, Sigonella will disappear.