It's impossible because of laws of thermodynamic. Star can heat it surroundings up to temperature of it surface, and with distance this temperature drops in proportion to square of distance. For example, Sun surface has temperature of 5800K, but it can heat surface at Earth only up to 450K, because surface of sphere at Earth orbit is much much larger than surface of Sun. Interstellar light cannot heat for more than fraction of Kelvin.

You're forgetting to sum over all the visible stars in the sky, which just brings us back to Olber's Paradox.

Think of it another way—if you're in an oven at 5800K, there is no arrangement of matter you can put inside that will not eventually come to thermal equilibrium.

You're forgetting to divide that sum by distances to these stars in square. Thermal equilibrium inside oven and in few light years outside of oven are two very different things.