The oldest library in the world no longer exists. It was built in 300 BC and located in the ancient city of Alexandria, Egypt. During the Alexandrian War, Julius Caesar accidently burned it down.
The oldest continuously running library is located in a monastery in Saini, Eqypt. However, St. Catherine's Monastery is not open to the public.
That leads me to the oldest continuous library open to the public: Bibliothèque nationale de France. According to the BnF website: The BnF collections are unique in the world: they include14 million books and magazines but also manuscripts, prints, photographs, maps and plans, scores, coins, medals, sound, video and multimedia documents, sets, costumes... These collections continue to grow regularly.
You see that this is quite an impressive collection. However, the buildings are just as impressive. In 1988, then president François Mitterrand announced the construction and expansion of what would be the most modern library in the world. Dedicated in 1996 after huge budget overages, it contains ancient texts, all publications ever printed in the country of France, and continuous art and manuscript exhibits.