It's a "Special Duty Assignment" that you have to apply for. The Army Old Guard, stationed at Ft. Myer in Arlington, is the oldest single unit in any of the armed forces, and it is, indeed, a great privilege to be chosen for it. They are the ones that guard the tomb of the unknown soldier in Arlington National Cemetary (which you can see out the windows of their dorm on Ft. Myer). As to the other services, each has their own "Honor Guard", and those are similarly difficult to get into.
Before you ask, no, I wasn't in either....While in the Air Force, I was stationed at the Pentagon, and lived on Ft. Myer in the same dorm (different floor) as some of the Old Guard guys.