I fiddled with CRTs and am still here (with minor scars) so it is possible. However a bare electron gun is pretty much useless except maybe as a cool sculpture component.
In order to work it needs a rather high 2nd anode voltage, of the order of a few kV. Old TVs (monochrome) tended to use about 15kV if memory serves, and more for color TVs (up to about 30kV). Electrons don't just shoot out of the end at high velocity like a 9x19mm round from the barrel of a Glock, they are accelerated by the high voltage toward the screen. Further, if you power up the filament in air it will last only minutes or seconds before oxidizing and failing.
The high voltage is by far the easy part. To get it to work it has to be in quite a good vacuum, which would require something like a two-stage mechanical roughing pump and a diffusion pump (or a very expensive turbomolecular pump that will die if mistreated). High vacuum systems require a certain set of skills that has to be acquired. Once the system is pumped down, a 'getter' (something like a barium material heated by an external induction heater) is used to gobble up some of the remaining gas. That's what the shiny spot was on the neck of the tube. None of this is impossible but it takes a lot of care and commitment and a certain amount of cash even if stuff is scrounged or bought on eBay.
If you really want to fiddle with CRTs try to get ahold of an electrostatic-deflection oscilloscope or radar display tube from mid-last century- maybe 3" or 5" round with green P1 phosphor. You need to take care about the potentially lethal high voltage on the second anode, anode and focus electrodes. Implosion injury if the glass breaks is also possible. Deflection voltages are often pretty modest (the smaller the screen size and the longer the tube, the less voltage it takes to deflect the beam a certain distance), and a dual op-amp can drive them directly in some cases. Pick one that you can get a datasheet for, in a language that you are fluent in (or that has a translation- there are some ex-Soviet tubes out there). Lower anode voltages will reduce the required deflection voltage too, but there are limits (brightness and perhaps optimum focus will suffer). TV set CRTs used electromagnetic deflection which is much more limiting (but it can result in higher deflection angles and thus shorter tubes).
Edit: FYI, the below (from this site) is what an electron gun (triple type for color) looks like: