While it is true that the Vatican holds an enormous number of manuscripts that are forbidden in access to, and the parts that are open to research are accessible only to a few researchers, picked by the Vatican itself, mainly because of their political and religious opinions (not only classics material but also early-Christianity and Judaism related material and historical manuscripts of the Vatican itself), this HARDLY stops research in any of these fields, including Classics. To call Classics a dead field of study is nothing short of ignorance.
While it is true that we are limited in our access to the Vatican archives, the desert, however, gives us many many evidences the Vatican archives can never hope to contain.
Papyrology is one of the most important fields of research in modern Classics (pardon the pun) research, and it gives us completely new works, long thought to have been lost, and much more accurate versions of known works. This is achieved by the fact that the dry air of the Sahara serves much better than the humid, moss-afflicted libraries of European monasteries in preserving paper (or papyrus), and therefore the works we have on papyrus are much earlier than European tomes, and therefore have gone through less copying, and their contents have not been distorted by various copying mistakes.
For instance, a copy I have of the Poetica by Aristotle in the original Greek, and that was edited in the 1960s is considered today completely outdated, due to numerous corrections done to it based on Papyri.
In addition, hundreds of inscriptions on walls, tombs, official buildings etc. are discovered in archaeological excavations every year. This is full of historical information. For instance a senatus consultum discovered recently in France helped shed some light on the murder of Germanicus.
While I, like any other classicist, would love to get our hands on the hidden treasures of the Vatican, to say that lack of access to them has brought Classics research to a halt is completely false.