In general, even the risk of condensation makes this idea impractical, at least on first inspection.
With water-cooling and "phase-change" refrigeration systems, at least the cooling is confined to water-blocks or coolant-blocks around the CPU, GPU and NorthBridge. Why does that restriction of low temperatures work to our advantage? Because we can use neoprene rubber grommets to insulate those particular parts against even a hint of condensation forming.
Two water-cooling ideas have been implemented by enthusiasts that may seem more radical than even external "Reserator" cooling reservoirs. One guy used an office water-cooler for his reservoir, so he could certainly be said to fall in the category of "chilled water cooling". Another guy buried a simple water-cooling reservoir deep in the earth of his back yard -- in a spot that was mostly under shade much of the time. His water was cooled to a constant 55F. I'm only speculating that there are less concerns about dew-point and condensation in this latter system.
Some water cooling systems put a TEC cooling-plate in the cooling loop, providing chilled water to the CPU waterblock and using a second water-cooling loop to route the TEC's heat to a radiator-fan assembly.
Using a dessicant to absorb moisture means that eventually the dessicant needs to be replaced. I have heard echoes of the idea where someone used a small "room air-conditioner" that also had a built-in dryer to push cool, dry air into the computer. Certainly, condensation problems would probably not plague that arrangement.
In the refrigerated system you imagine, it would have to be filled with absolutely dry air and sealed against water-vapor contamination. This presents other maintenance "access" problems.