In the absence of convection and conduction, you might be tempted to assume that any exposure to radiation would necessarily raise the temperature of an object over time, as it's receiving energy all the time.
But that's missing a step. Every macroscopic object also emits radiation based on its temperature. We don't normally think of this because everyday objects emit infrared radiation that we can't see, and at low enough levels that it doesn't raise our temperature and thus we can't feel it either. But I'm sure you've seen (at least pictures of) metal heated up enough to glow; it actually "glows" even when it's cold, just in infrared where you can't see it.
So every object is both absorbing energy from radiation and emitting it, all the time. It will heat up over time if it is absorbing more energy through radiation than it is emitting, and cool down over time if it is emitting more energy than absorbing. But as an object gets hotter it emits more radiation, and as it gets colder it emits less. If you wait long enough in a constant environment it will eventually reach equilibrium (where it is absorbing and emitting exactly the same amount of energy), and stay at that temperature forever.
The cosmic background radiation is about equivalent to the thermal radiation you'd see if you were surrounded by a shell at 2.7 Kelvin. That means that any object you place in space that is warmer than 2.7 K will be losing more energy in the radiation it emits than it will be absorbing from the cosmic background; it will thus cool down. Only an object that was already colder than 2.7 K would actually be warmed up by the cosmic background, and only until it reaches 2.7 K.
