Each TV has a specific distance (or range of distances) where the screen looks its’ best. This is based on the size of the screen, and the density of the pixels on the screen.
Sitting too close will cause you to notice the individual pixels, rather than having them blend into a smooth picture. It looks similar to the blocky, granulated appearance you get from expanding a digital photo too much.
But sitting too close like this doesn’t really have any harmful effects on your health, though it can cause you to get a headache.
I don’t know enough to comment about the possibility of harmful radiation from old TV sets. Somebody else will have to address that.