That looking too closely at the screen will blind you or damage your eyes. This myth originated decades ago in the 1960s from an advertisement by a television manufacturer. Basically in 1967 General Electric reported that their color TVs were emitting too many x-rays due to a factory error, so health officials recommended keeping children and pretty much anyone else at a safe distance from the screen. The problem was soon resolved, but the myth endured.
If you ask me I would say that x-ray radiation has little to do with going blind, I have no idea if radiation can actually make you blind, but it's funny how somehow eye diseases got in the way as the only possible consequences in the myth just because we use our eyes to watch TV.
CRT screens generate bremsstrahlung (x-rays) from slowing electrons, so the front piece of glass is normally made of lead glass, or barium-strontium glass to block it.
After the General Electric incident, testing showed that nearly every manufacturers TVs were emitting too many x-rays. This led to the recommendation to stay 6 ft. away from the TV when it was on.
The FDA then later imposed limits on how much radiation a TV was allowed to emit.
With the these regulations, if you were to absorb all x-rays from a CRT for 2 hours a day, every day, you would get 320 millirem per year (comparable to the average US background radiation of 310 millirem per year).
See here, as well as this article.
Edit: Also, significant doses of x-rays can blind you. Radiologists in medicine particularly have to shield against it, since they are exposed to it every day, and exposure builds up. See here and here.
Edit again: Wasn't paying enough attention. That last source talks about ionizing radiation specifically, so not x-rays.
Mmm. I worked on CRT screens when I was in the US Navy and had some CRT monitors in the past.
After a long session, my eyeballs 100% felt 'burnt' inside.