Generally the real issue with cooling systems is getting rid of waste heat. A typical cryocooler today is a Stirling Engine being externally powered, so you have an engine attached to the cryocooler, which is generating heat, plus the cryocooler itself, which is rejecting large amounts of heat to bring the temperature down.
So you will be wearing a backpack with an engine, fuel or energy source, cryocooler (and presumably associated systems for controlling the rig, passing coolant to the nanomachines etc., all of which are radiating heat. You will need some sort of radiator mechanism to reject the waste heat, and do so in a controlled manner so you don't fry yourself or interfere with the production of the cryogenic coolant either. So your backpack may be sprouting a set of "wings" to act as radiators, something like Polish Hussars. While fairly spectacular as a visual, it also means the enemy can both see you in infra red and thermal imagery, and has a very conspicuous target to aim for in a fight.
Aim at the wings...aim at the wings....
While future technology may change this, the other thing which needs to be taken into account is that generally speaking, larger systems are more economical and efficient than very small ones. While it may indeed be possible to have a backpack unit, a building sized unit for fixed operations will be able to do the job with less energy per unit, and possibly handle far mor nanomachines per unit energy than a backpack.