Is it dangerous to run a CPU in an extremely cold environment?
As long as you keep the machine dry, there shouldn't be any real problems from running a PC in the cold. Where you will run into problems is if condensation is allowed to form on the components.
I have installed computers into blast chillers and refrigeration factories without problems, but we did need to make sure the cases were put into enclosures that keep moisture out.
We actually purchased special electronics cabinets for that purpose, but its not expressly necessary as long as you can keep the moisture level under control.
It may not work stable when temperature is too low because electrolytic capacitors did not work properly when temperature is too low. Especially when the power supply is cheapest possible (with small capacitors).
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.
Most limitations can be traced to the electrolyte. At high temperature, the water can be lost to evaporation, and the capacitor (especially the small sizes) may leak outright. At low temperatures, the conductance of the salts declines, raising the ESR, and the increase in the electrolyte's surface tension can cause reduced contact with the dielectric.
The conductance of electrolytes generally has a very high temperature coefficient, +2%/°C is typical, depending on size. The electrolyte, particularly if degraded, is implicated in various reliability issues as well.
Source: Wikipedia: Electrolytic capacitor -> Reliability and length of life
This is why only CPUs (and other semiconductor components) are cooled to low temperatures, not the whole computer.
The normal working range for most capacitors is -30°C to +125°C
Source: Electronics Tutorials: Electronics Tutorial about Capacitor Characteristics
As it happens, there's a bunch of experimental servers running quite close to my office — specifically, on the roof of the building I work in — that may provide some relevant real-world data.
They're part of an experiment by the Helsinki Institute for Information Technology on cooling servers with unfiltered outdoor air at ambient temperatures ranging from around -20°C to +20°C. Originally, the servers were simply placed in an off-the-shelf camping tent (to protect them from snow, rain and pigeon droppings) and left to run over the winter. Since then, they've built a small rooftop greenhouse to run the servers (and grow some plants) in.
So far, the experiment seems to be a success. As one of the researchers writes,
"The intake server air is not filtered in any way. The most frequently asked question is, whether the moisture condensing inside the servers harms them. However, the condensation works in reverse, since the servers are cooled by air, so they are always warmer than the air around them."
If we use a machine in here, will it be at risk of damage from the cold? Is there any considerations we need to make?
Consider a hot summer's day, and you've just finished cutting the grass in your backyard. You go inside and bring out a cold beverage, sit down, and enjoy the landscape... That's when you notice that your drink is covered in condensation.
Consider a cold winter's day, you've just finished shoveling off your driveway. You go inside and bring out a hot beverage, sit down, and enjoy the landscape... That's when you notice, well, nothing - you won't have any condensation formed.
This is why in most cases, simply exposing a computer to the cold won't do any damage. Computers enjoy being cold rather than hot, and so long as you're not moving a cold computer into a hot room (like the first example, this could cause condensation inside the case and thus short-circuit some components), there shouldn't be a huge issue.
Do note that if you ever have to bring a freezing cold computer back indoors (i.e. you stored the computer outside at -10C, and bring it inside at 21C), you should make sure all of the condensation has been cleared before plugging it in and starting it up. If you're unsure, I would let the computer sit inside for a few days for any moisture to evaporate, and for the temperature/humidity in the case equalize.
Lastly, also ensure that if the computer is exposed to the weather elements (i.e. precipitation, high humidity, etc...), additional precautions/safeguards might need to be taken (waterproof case, moisture-proof case, proper ventilation).