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.