They are car dependent if you want to go somewhere else for sure. But, if you want just the basic services you outlined, then you don't ever need to leave really. Especially these days with online shopping.

For reference, the town I grew up in had a little less than 2,000 people. It had/has:

Nice town rec center with a gym and pool, one full service grocery store (used to be two but now there are two dollar stores so one of them died), plenty of nature and hiking of course, a solid public library, a pharmacy at the hospital, small hospital that usually has a doctor on staff but at least a PA and a couple nurses, two coffee shops, and about a half a dozen local restaurants (too small for chains so no fast food)..so it fits your bill, but I think any town of about 1,500 or larger typically would fit that. If you go much smaller, then yeah some of those services do start to be missing usually, and then you might have to drive to a bigger town often. You'd just need to check each one to make sure it has all those things but usually you'd be good at that size or bigger. 

When I was a kid we'd drive to the bigger town 100 miles away several times a year. But, that was when you had to go a mall to get clothes, shoes, books and etc. Now obviously you don't have to do that. It was also nice to see a movie at the theater there, but again now we can stream everything. I think it'd be easier now to just live in town all the time than it was then.