The National Film and Television Archive, which is part of the British Film Institute, is the repository for a lot of the old BBC tapes and kinescopes. Viewings can be arranged, but not copies.
The BBC still owns the rights to its programs. It will not make copies for personal use.
BBC Research Central FAQs
The
BBC Shop does sell tapes of some old BBC shows, including some sports events. Its online shop has a motor racing section, but it is empty. You could write to them and suggest that they produce a tape of old motor racing broadcasts, but don't hold your breath while you wait for a reply. I know some people who have written regularly, asking for the Donald Campbell drama
Across the Lake, starring Anthony Hopkins. As yet, no video, no reply.