Is it safe to run computers for long periods of time in very cold environments?
Operation
There are several issues with operation of electronic equipment in low temperatures:
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Sudden change of temperature causes mechanical stress as components thaw and freeze again repeatedly. This can cause electrical contacts to weaken, heatsinks to work loose, etc.
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Some components simply do not work or perform poorly in low temperature: electrolytic capacitors and other components change their properties; LCDs simply would not work in sub-zero temperatures, etc. Wikipedia says that:
Additionally, low temperature is a problem for most aluminum capacitors: for most types, capacitance falls off rapidly below room temperature while dissipation factor can be ten times higher at −25 °C than at 25 °C
Military-grade laptops cost a fortune for a reason... This being said, I have an old computer in my own unheated garage, and it survived occasional use in Canadian winter for two years without a problem. I keep its cover removed to lessen a chance of condensation, don't know if it is feasible for you since you drive a CNC with it (I use it for 3D printer so dust is not an issue).
Some say it's better to keep the cover off, so there is no sharp temperature change when the computer is turned on, some say it's better to keep the cover, so the internal components heat up a bit. I know that some Wi-Fi routers designed to work outdoors are actually heated...
Storage
I think LCD minimum storage temperature is usually above -25C, that means it can cause permanent damage (but check your manual). Otherwise computer should be ok, as long as you thaw it gradually to prevent condensation.
The hard drives "breathe" more under these conditions if the drive is allowed to go through warming and cooling cycles. In the Pacific Northwest you can end up with moisture condensing out inside the drive housing. We keep computers running 24/7 inside a cabinet where they can keep it relatively warm. Lower temperatures hopefully alleviate this kind of thing with less moisture being in the air.
Also with the new lead-free solders, surface mount chips, the high tin content can lead to problems under extreme cold. It actually deteriorates. We had the cold pop a few chips on one of our HP printers controller causing it to be warrantied. The service tech told us that it was due to the lead-free solder having less flexibility.