The pub dates to the 17th century and doesn't seem too much changed since then (except for electricity!). The rooms are low-ceilinged, dark, and small. The reason it's famous is that it was a gathering place for the famous writers' group called The Inklings, which included J. R. R. Tolkien, C. S. Lewis, Charles Williams, and others. They used to meet in a back room here for lunch on Mondays or Tuesdays, and read their works in progress to one another. (Inspector Morse in the novels used to come in here a lot, also.) When we were there it was very crowded. Probably most of the people in there were Tolkien fans on a literary pilgrimage :-)
On my last free afternoon, I was walking by the Sheldonian and saw that it was open. The other times I'd been by there, it was closed. The building is basically a large auditorium.
When I paid my admission they gave me a booklet describing the degree granting ceremony that takes place here. Typical of Oxford, I don't think it's changed in centuries. There is a lot of processing back and forth, and bowing, and coming in one door and out another, and speeches according to a strict script--all in Latin! I can imagine the students thinking: "Is it over yet? Do I have my degree yet, or what?"