Sadly no. Space is cold but also very low on convenient mass to transfer the heat into. Vacuum is an ideal insulator. On Earth, you can put the heat into water, air, thermal pastes, etc., but in space, you have only slow old black body radiation, unless you do something more creative.
Space in itself has no temperature. It does have some background radiation, and thus you can measure its temperature (pretty cold at something slightly over absolute zero).
Here's the thing though. There's very little matter even in low earth orbit, so you can't use matter to transfer heat, as you would do on Earth (conduction and convection are out). That leaves only radiation, which requires a pretty large surface area. Try placing a computer on vacuum even on Earth and report back how the temperature looks like.
So no. Getting rid of heat is a big issue over there. Even in the shade.
Not as well as intuition suggests. Deep space is cold, but also the sun (and reflected energy from the Earth!) is pretty hot. Without air, there's no convection so it's all radiated energy transfer, so you're limited by how much surface area you can "point at" deep space while not also pointing it at the sun.
Space near Earth is hot (apparently, ~ 120°C) in the sun, and cold in the shade (e.g., the Earth's shadow). On average, it's not particularly cold. You can get colder average temperatures on Earth, without the wide swings.
But more importantly, space is empty, and the best way to cool is to dump heat in some medium which carries it away (air is the basic one on Earth, but applications that really need cooling like to use water as a primary medium.) In space you've got...nothing, basically, somyou are stuck with radiating, which works, but poorly.
It's not that heat can't go anywhere, it's that there's nothing to carry heat away.
On Earth, air molecules can carry heat away from you. You transfer heat to the air around you. In space, there is almost no matter around you to absorb the heat and carry it away from you.
Your only option is to radiate heat away from you in the form of infrared light but that is a slow process.
Vacuums (or near vacuums) are actually good insulators. Double-walled insulated travel mugs are a great example of this.
Also, if you think about this in the context of the Sun, you get a really visceral feel for just how much energy is being produced. Black-body alone is transferring that energy to us. Scary.