The electron speed in CRTs is close to c.
"Typical kinetic energies coming off the cathode are on the order of 0.1 eV. After acceleration typical kinetic energies are 15-25 kEv, depending on the size of the CRT. The corresponding speeds are much less than c."
http://www.gatago.com/sci/physics/relativity/14203734.html
electron beams dissipate *very* quickly in air, you won't hit anything with your gun.
that won't be a problem, as the electrons only need to "hit" a magnetic field.