Paths to specialize in antique book and manuscript conservation and handling I am not sure whether this is the best place to field this question but I assume Academia is a reasonable fit. If someone is interested, initially as a hobby, in the collecting, cataloguing, rebinding, and conservation of old books and manuscripts what would be the best path to pursuing those activities in a more professional venue? How do people specialize in book handling and conservation? 

From a hobbyist perspective say someone collects old Scandinavian Literature over 100-200 years old and engages in the study, preservation, and cataloguing of such books for posterity (these days one may do this individually with such digital online archiving projects as Project Gutenberg and Archive.org - sometimes accompanied with OCR for easy indexing of the text online). If such a person was interested in doing that more often in some formal capacity how would they pursue that? 

Does one have to study Conservation Science and obtain a degree to handle documents for institutions that have rare and antique book collections, or say for museums that need periodic restoration of books from page yellowing and foxing? Could one alternately volunteer for a library that has a rare books department and enter in from that avenue? How does anyone end up being a manuscript conservation specialist? Surely the U.S. Library of Congress has people who do such things. And what title would they typically hold?
Book conservation, restoration, and preservation are parts of library and information science. I suggest that you look into accredited masters programs in that field. In the US, the American Library Association is the accrediting body.