https://id.loc.gov/download/ offers bulk downloads of many of the vocabularies, including the subject headings. In addition, individual records of various types can be search for.
That seems to have Subject Headings, but not the Classification, unless I'm blind (a fairly high likelihood).
I've spent quite some time poking around the LoC website over the years without finding anything more accessible than the PDFs of the Library of Congress Classification. Are you aware of that being generally available?
Near as I can tell, the best that's available is starting with the class(es) you want (e.g., https://id.loc.gov/authorities/classification/H.html for social sciences) and crawling recursively to grab whatever serialization of the RDF you want to consume. There's no SPARQL endpoint for id.loc.gov, alas, but it beats resorting to the PDFs.
