if the Sun were 'shut off', Earth's surface temperature would drop by a factor of two about every two months. The current mean temperature of the Earth's surface is about 27 °C / 81 °F. This means in two months the temperature would drop to -123 °C / -190 °F, and -189 °C / -324 °F in four months. So basically except some bacteria seem to be capable of surviving at extremely cold temperatures in space, we humans would die pretty quickly.