CRTs
For a colour monitor, don't forget the shadow mask.
For early generation monochrome monitors, there used to be an offset bias on the beam deflector so that the beam did not strike the phosphor at right angles, but at an angle that would aim the beam away from someone sitting directly in front of the monitor.
Electrons from an electron gun in a CRT are relatively low energy, and can easily be stopped by the metalised inside coating of the glass, and the glass itself. And the energy was not high enough to generate X or gamma rays.