Out in town the houses really don't have yards. You'll usually have a parking space (1 that maybe fits 2 kei cars--yellow plated mini cars, or 1 regular car). You'll have a walkway around the house made of gravel. No grass, no place for large dogs to run around.
Everyone I know that has dogs lives onbase. When you get here you'll rate a garden apartment that's 2 bedrooms. The units are 2 upstairs 2 down. The downstairs units have a small lawn off the back. I know of 1 couple that has a lab, and they live in a top unit. She does fine, but they walk her a lot...the wife is a dentist on base and is able to be home at lunch to take the dog out.
Attached is a pic of the back of the units, you can get an idea of the back yards. The side yards belong to the upstairs units. I have seen people keep their dogs in those (but there is no access from the unit itself).
The only reason I said that I would live off base if we came back was the fact that we haven't yet. I like the apartment on base, there's enough room for 2 people (actually 1, the husband is gone A LOT), and it's close to everything. My husband rides his bike to work, and when he's on duty he doesn't have to stay in the Q. The only bad thing about living on base is it's a small community so everyone knows your business. On the other hand, it's nice to have people so close that can relate to what you're going through. In our building we have 1 couple from the same squadron that live downstairs, and it's nice to just pop down and have dinner, etc. A new couple from HSL just moved in next door and her husband is on the small boy det with mine, and this is her first deployment so I've been helping her get stuff settled and informed about how things work.
There are 2 sides to it, I know my friends who live out in town love it, but also have issues like we do on base.
It's all about what fits you guys. If you want more pictures, or info just let me know (about Atsugi, or even HS14).
(see, I can go on and on...)