The Steven F. Udvar-Hazy Center has a space shuttle but you can't go inside it. Are there any that you can or are there any that you can under special circumstances or something?
An exhibit featuring the official high-fidelity replica, latterly dubbed Shuttle Independence (called Explorer when it was in Florida), atop the 747 shuttle carrier aircraft, is currently under construction at the Johnson Space Center in Houston and expected to open in February 2015. Currently, plans call for visitors to be able to board the SCA and enjoy "an unforgettable experience aboard the full-scale shuttle model," although details are not available.
Until that exhibit is complete, your next best bet is the brand-new Space Shuttle Exhibit at the National Museum of the U.S. Air Force near Dayton, Ohio, which features the Crew Compartment Trainer, a life-sized mockup. The exhibit "allows visitors to experience the size and shape of an actual space shuttle orbiter by entering the payload bay and looking into the flight deck and mid-deck levels."
My third suggestion would be the Full Fuselage Trainer on display at the Museum of Flight in Seattle, Washington. It does not have the wings, but as its name indicates is a full-sized mockup of the fuselage that was used for astronaut training purposes. Tours of the interior are available for a fee on weekends and public holidays.
Shuttle Inspiration, a life-sized mockup of wood and plastic built by Rockwell in 1972 for its official proposal to build the shuttle, is now owned by the City of Downey, California. There are plans to create a museum around it, but it is currently disassembled and in storage until funds can be allocated. I remember climbing on it on a science day for the Boy Scouts way back in the 1980s; it gives you a sense of its size and the scale of the crew compartments, but of course there wouldn't be any genuine artifacts, making it just slightly better than various other independently created replicas that seem to be floating around.
The four actual orbiters on display, as you know, cannot be entered by the public:
The Kansas Cosmosphere and Space Center in Hutchinson, Kansas is the largest Space Exploration museum in the world, and has a full-size space shuttle on display--actually built into the building, you'll see it when you first enter. If memory serves, it never flew, but was built as part of the shuttle program. You cannot go into it, but you can see it pretty close up.
They have many other artifacts, including real V1 and V2 rockets, the recovered Apollo 13 command module, and a few artifacts that you can go inside (although they may be replicas).
So sadly, you cannot go inside of the Space Shuttle here, but if that's your sort of thing, this museum gets pretty close, and would still be a must-see on your list.
Further note that going into a space shuttle probably is never practical, unless you work for NASA, as all space craft are built to be as compact as possible, making them practically impossible to open to the public. They aren't built like ships or airplanes, designed for people to walk on board, and move around comfortably. The portions intended to hold people are either tightly-confined seats, with full harnesses, used during take-off and landing, or are long, tube-like structures, intended for use in zero gravity.
I have been in a space shuttle simulator, used at Spar Aerospace by the astronauts who flew the shuttle (they were practicing with the Canadarm) so I decided to see if any of the simulators are still around.
One seems to be headed for Texas:
That university has several pages with similar wordings but less and less confident dates, so perhaps until they raise the money it's not really there yet? However they definitely intended it to be open to the public:
Two are in place and ready to be visited in Dayton, Ohio:
So those are interesting but not only didn't go to space, they weren't used by astronauts.
Finally, Spar still sort of exists, as MD Robotics, a subsidiary of the MDA Space Missions division of MacDonald Dettwiler. They still have a facility in Brampton so it's possible that a tour can be arranged. (Mine was because my mother worked for Spar in software QA on the arm.)