Although the previous posts are mostly very good, here are a couple of additional points. First is that over the past 50+ years, TV shows have been shot on dozens, if not hundreds, of different film stocks with widely varying characteristics. So obviously there is no one answer to the OP.
Even in the 1950s and '60s, 35mm motion picture stocks had resolving power well beyond HD quality. After all, in theaters those images were projected to 60 or 70 feet wide, and appeared sharp and clear. Furthermore, Kodak and other film manufacturers have not stood still in the last three decades. Believe it or not, in the last decade or so, film image technology has advanced more rapidly than digital imaging technology in terms of resolution and low-light sensitivity.
So theoretically, any TV show shot on film should be capable of a very high quality HD release.
However, there is always a difference between theory and reality. The primary issue, especially for older shows, is the availability of the original film elements and their condition. I would guess that for many, if not most, shows from the 1970s and earlier, it would be rare to find well preserved film elements for the majority of the series’ episodes. After the shows were transferred to video for broadcast, production companies had little reason to save the film, since there was no inkling then that home video – to say nothing of the Internets – would become a new revenue source. So I imagine the vast majority of that material is long gone.
In many cases, the original video masters are also probably lost or in bad shape. I would guess that the DVDs of most 1960s and '70s shows available now were mastered from tapes that are several generations down from the edited film master.
Since the advent of VHS home video, DVDs, and the Web, producers have recognized the huge sums of money available from ancillary sources and have therefore been more protective of their assets. So it’s generally easier to make high quality transfers of those shows, because more of the original elements have been saved.
Finally, it is now possible with digital technology to significantly enhance images, removing noise and increasing resolution. Imax Corporation is able to take 35mm Hollywood films and digitally blow them up, virtually seamlessly, for 60x80-foot IMAX screens. This level of enhancement is far beyond what would be needed for HD. Of course, it’s very expensive to go to that extreme, but a similar process for HD resolution would be cheaper. It’s just a question of whether DVD distributors will find it necessary and worth the expense.