If you're single and live in barracks or on ship - no pets. If you're single and live on the economy, pets are your choice. As HYNChick pointed out, landlords are not always accepting of pets.
Some other considerations:
1) What happens to your pet on your duty day? You're required to stay overnight at your unit X number of nights per week as on-call assistance, where's your pet?
2) What if you get temporary orders to a school or another unit for 1-4 months? This does happen as a matter of routine to many ratings.
3) When you finally get tour complete and are up for transfer, what happens if you get assigned to a ship? No pets allowed.
Just some things to consider.
Disclaimer: I'm not a doctor or a recruiter. Verify any and all information I've stated before making a fool of yourself.