Yes, it is possible to have a "cold" solar system or a system of planets without a star. Such systems are known as rogue planets or free-floating planets.
Rogue planets are planetary-mass objects that do not orbit a star and instead wander through space on their own. They can form through various mechanisms, such as gravitational interactions within a planetary system or through ejection from their original star system. These planets can be as small as Earth or as large as Jupiter, and they can have atmospheres, moons, and other planetary features.
In the absence of a nearby star, these rogue planets do not receive significant amounts of light or heat. As a result, their temperatures can be extremely cold, approaching the temperature of interstellar space. However, it's worth noting that even in the coldness of space, objects can still retain some residual heat from their formation or from internal radioactive decay processes.
Rogue planets can be detected through various methods, such as direct imaging, gravitational microlensing, or their effect on nearby objects. While their exact prevalence in the universe is still uncertain, astronomers have already discovered a few candidate rogue planets in our own galaxy.
It's important to distinguish rogue planets from planets within a planetary system that lack a star. Rogue planets are not associated with any star, whereas planets within a starless system would still orbit a central gravitational center of mass but not a conventional star.