Nope, you can't go in between the stones. They basically have a fence around the outside of the large circle.
I went to Stonehenge as part of a tour, so we were there for maybe 2 hours...which was plenty of time...unless you were really interested in all of the different theories about its uses and creation...but still I'd say it's at most a half day trip.
We did go to Salisbury for the afternoon...which was less than thrilling...but then we travelled on to Oxford, where we spent the next day (which I loved!).