The short answer to the question is: in museums and libraries throughout the world.
The sheer number of fragments alone means that no one academic entity - let alone even one government or ecclesiastical authority - can "own" them all.
There are over 5500 manuscripts, miniscules, unicals, papyri, parchments, and fragments that critical scholars have used to ascertain the authentcity of the Scriptures as we have them today. A corpus like the Nestle-Aland 28 brings together some of the best consensus from the history of the manuscripts to most definitively decide what the "original" text most likely said - and the use of an appartus will illustrate the various manuscripts that were used to make the determination.
For more about apparatuses and manuscript evidence, you should check out other questions already here.
With over 5500 manuscripts, then, a complete list is too broad. That said, there are a few key manuscripts that scholars would go to right away:
Dead Sea Scrolls, at the Shrine of the Book at the Israel Museum in Jerusalem. Side note: I've been there, and it is just way cool. The museum itself looks like one of the jars in which the scrolls were found, and it showcases the Great Scroll of Isaiah. The actual scrolls themselves can be viewed online.
Codex Vaticanus, at the Vatican Library in Rome. Did you really thing the Vatican wouldn't have one of the best manuscripts in its own library? This too, is online, but trust me, it is worth the visit.
Codex Sinaticus, at the British Museum and three other museums: the British Library, the Library of the University of Leipzig, the National Library of Russia in Saint Petersburg, and the Holy Monastery of the God-Trodden Mount Sinai (Saint Catherine’s). These were distributed for "safekeeping", after being "rescued" (sounds so much better than "stolen", eh?) from St. Catherine's Monastery in the Sinai, by Tischendorf, back in 1859. It is available online for viewing. Sadly, while I have been to the British Museum, I didn't see this! (Too busy looking at the Rosetta Stone, and the Cyrus Cylinder which may be the oldest extra biblical verification of a bible text, dating from the 500s BC.)
Interestingly, the oldest known fragment of the New Testament - p52, John Ryland's fragment of a verse in John - is located in library at the University of Manchester, England.