In this particular case, we can establish the temperature of space quite satisfactorily, the very same way we establish it here down on Earth: by sticking a black body analog in the volume we want to examine and assessing the temperature this object reaches at the eventual equilibrium - in shadow, and in sunlight, for two very different readings.
So let's stick a thermometer in space at the distance of one AU from the Sun, that is, in the vicinity of Earth. In shadow, we get something close to 70 Kelvin, or minus 200 degrees Celsius. In sunlight, we get about plus 120 degrees Celsius. Not too bad in terms of survival; many planetary environments are more extreme.
But the Valiant
marker would probably have reached an equilibrium more than 1 AU away from the nearest star. Delta Vega's sun was said to be a few lightdays away; unless she were really hot, temperatures at the Barrier would probably be in the order of just a few dozen Kelvin, enough to liquefy the air around the recorder marker when it got aboard unless some adjusting were done. The piece of the Charybdis
beamed aboard explicitly came from a place with very low temperatures (indeed, below zero Kelvin was quoted!), again dangerously cold by any standards. Objects floating in open space near stars would probably be constantly exposed to local sunlight and might actually be hot to the touch, but I don't think any such were encountered in Star Trek, as the heroes and villains would typically only go near stars when also going near the local planets... In practice, everything interesting and worth picking up would be found on planetary orbits.