It's been interesting to see how TV shows from the 60s and 70s are being made available in HDTV formats. I've watched a few of Classic Star Trek, where they not only rescanned the old film at better resolution, but also created new computer graphics to replace the old 60s-era opticals. (Oddly, because the relative budget for these graphics is small, some of the graphics look a bit cheesy in a different way, even though much higher in technical quality.)
The earliest TV was shot live. My mother was a TV star in the 50s and 60s, but this was before videotape was cheap. Her shows all were done live, and the only recording was a Kinescope -- a film shot off the TV monitor. These kinneys are low quality and often blown out. The higher budget shows were all shot and edited on film, and can all be turned into HD. Then broadcast quality videotape got cheap enough that cheaper shows, and then even expensive shows began being shot on it. This period will be known in the future as a strange resolution "dark ages" when the quality of the recordings dropped. No doubt they will find today's HD recordings low-res as well, and many productions are now being shot on "4K" cameras which have about 8 megapixels.
But I predict the future holds a surprise for us. We can't do it yet, but I imagine software will arise that will be able to take old, low quality videos and turn them into some thing better. They will do this by actually modeling the scenes that were shot to create higher-resolution images and models of all the things which appear in the scene. In order to do this, it will be necessary that everything move. Either it has to move (as people do) or the camera must pan over it. In some cases having multiple camera views may help.
When an object moves against a video camera, it is possible to capture a static image of it in sub-pixel resolution. That's because the multiple frames can be combined to generate more information than is visible in any one frame. A video taken with a low-res camera that slowly pans over an object (in both dimensions) can produce a hi-res still. In addition, for most TV shows, a variety of production stills are also taken at high resolution, and from a variety of angles. They are taken for publicity, and also for continuity. If these exist, it makes the situation even easier.