I would say X-ray laser is possible, but just too electrically hungry and huge to be useful to hobbyists, but let's say they make down to the semiconductor laser size, the energy bandgap makes it almost impractical (I would imagine it could ask for 18 - 22 Volts or so basing on III-V semiconductor bandgaps).
But having studied upon X-ray optical physics, it is kind of a bad news in itself, in partly because you will NEVER see the X-ray laser radiation, because it's obviously invisible unless it's in several hundred milliwatts or so, which then Oxygen would get excited enough to make laser light visible - it's not the visibility or lack of, it's also the Compton scattering you should be worried about (as X-ray laser radiation tend to be far worse than X-rays from the dentist tube). It also tend to cause damages much sooner owing to the property of the light (5mW green laser light can be seen as brighter than 5mW light from LED due to coherency of the emissions so localized heating can be magnified when the optical wattage goes up).
Lastly, optical setup is very difficult to deal with - you must be good at mathematics so you can predict where the light strikes, to achieve what is desired because mirrors and regular glass lens won't work at all (glass tend to be opaque in the X-ray wavelengths).
Nevertheless, having a desire to have X-ray laser is not exactly healthy. I originally wanted to build one until I figured out how much it will cost me (ie. pulsed discharge Argon / Helium callipary tube laser which tend to use very special capacitors which is also sometimes used in the larger multi-megawatts flashlamp-pumped Nd : YAG / Ruby lasers) so I dropped it. What you can't see is not something you want to use, especially when you start to smell the ozone when it gets turned on. Not to mention radiation shielding which tend to be both expensive and cumbersome.
