Is an electron gun dangerous? Is an electron gun dangerous & what would be the repercussions to human tissue?
If I were to do a DIY electron gun and put my hand in the beam would I regret it?
[Edit] What sort of energy transfer over what distance in air could be expected?
 A: I'm not a professional, but from what I understand all these devices really do is produce an RF field, and unless it's gamma radiation or it's strong enough to burn you up (such as if you were put in a giant microwave oven) RF will not hurt you.
That said, I would not stick parts of my body in front of anything that emits any kind of radiation without a good reason.
A: An electron gun has a strict electronics definition

is an electrical component in some vacuum tubes that produces a narrow, collimated electron beam that has a precise kinetic energy. The largest use is in cathode ray tubes (CRTs), used in older television sets, computer displays, and oscilloscopes. They are also used in microwave linear beam vacuum tubes such as klystrons, inductive output tubes, travelling wave tubes, and gyrotrons, as well as in scientific instruments such as electron microscopes and particle accelerators

Bold mine.
Vacuum is the crucial component here, so the only way an electron gun can hurt you is by x-rays generated when it hits the television front or the valve anode etc. You cannot put your hand in the vacuum without destroying the tube.
It is also found as a weapon when the energy of the beam is high

A particle-beam weapon is a weaponized version of this technology. It accelerates charged particles (in most cases electrons, positrons, protons, or ionized atoms, but very advanced versions can use more exotic particles) to near-light speed and then shoots them at a target. These particles have tremendous kinetic energy which they impart to matter in the target’s surface, inducing near-instantaneous and catastrophic superheating.

All the above  either is  highly classified information or on the science fiction front.
In any case if one wanted a weapon made out of accelerating particles  the electrons are not a good candidate: they are absorbed fast in the atmosphere and it will be difficult to get them out of the vacuum where they must be created without boiling the intermediate level. As a weapon radiation has the best chance rather than charged particles.
I would not advise you to get into the beam line of a  weapon and anything with high enough energy to produced boiling and death at the target will also be producing a lot of X-rays so it would be unwise to handle it without robots.
A: "an electron gun" is not well defined.  If the energy of the electrons is low, they will stop quickly in air, so will not cause a problem.  If the energy is high, it depends what the flux is.  Low and high depend on how close you can put your hand.
A: An electron beam in air has the potential to be very lethal.
If you have any doubt, Search "electron beam sterilisation" on you tube. 
"How Does the Electron Beam Sterilisation Process Work?"
See here
E-beam sterilisation is used as alternative to sterilising medical objects with gamma rays from a cobalt60 steriliser. Only e-beams dose the boxes much more quickly than a cobalt60 steriliser. 1-2 Minutes as opposed to 6-12 hours.
See here
An exposure to a cobalt60 steriliser of only a 60 seconds killed a technician trying to clear a jammed box of medical items on a conveyor belt in Israel. (search "the radiological accident in soreq"). 
If an e-beam steriliser can do in one minute what a cobalt 60 steriliser does in 6 hours to medical items, I wouldn't fancy my chances putting any part of my body in the way of an e-beam from a steriliser unit.
A: Not sure about electrons, but a guy was hit in the head by a high energy proton beam. It caused serious damage, but he lived.

As a researcher at the Institute for High Energy Physics in Protvino,
  Bugorski worked with the largest Soviet particle accelerator, the U-70
  synchrotron.[2] On 13 July 1978, Bugorski was checking a
  malfunctioning piece of equipment when the safety mechanisms failed.
  Bugorski was leaning over the equipment when he stuck his head in the
  path of the 76 GeV proton beam. Reportedly, he saw a flash "brighter
  than a thousand suns" but did not feel any pain.1
The left half of Bugorski's face swelled up beyond
  recognition and, over the next several days, started peeling off,
  revealing the path that the proton beam (moving near the speed of
  light) had burned through parts of his face, his bone and the brain
  tissue underneath. As it was believed that he had received far in
  excess of a fatal dose of radiation, Bugorski was taken to a clinic in
  Moscow where the doctors could observe his expected demise. However,
  Bugorski survived and even completed his Ph.D. There was virtually no
  damage to his intellectual capacity, but the fatigue of mental work
  increased markedly.[2] Bugorski completely lost hearing in the left
  ear and only a constant, unpleasant internal noise remained. The left
  half of his face was paralyzed due to the destruction of nerves.1 He
  was able to function well, except for the fact that he had occasional
  complex partial seizures and rare tonic-clonic seizures.

