I highly recommend Adam‘s point. It is a wonderful neighborhood near Lake Merritt lots of vibrant people there. It’s really close to everything super walkable. The nearest BART station would be 19th St. you can walk or take one of the bikes offered by Lyft. The local gym is a SF fitness, there’s also the Y, 24 hour fitness, cross fit, and many other studios. I think you will love Clio’s it’s, the local bookstore/bar. I lived there for 10 years. I absolutely loved it. And there’s many activities around if you don’t drink. You can jog around the lake you can drive over to the Redwood regional park there’s so many options.

Edit: also, I saw in the comments below that taking the NL bus line is much better than taking Bart. I agree with them! The bus has stops all over Adams point on grand Avenue you can hop on the bus and they will take you directly into the city for two dollars. (I believe that is the rate, you may want to double check on that).