What would happen to life on Earth if the Sun were to burn out how long would we be able to survive?
The Earth's atmosphere has some capacity to hold in heat but not much of one. A relatively simple calculation would show that the Earth's surface temperature would drop by a factor of two about every two months if the Sun were shut off. The current mean temperature of the Earth's surface is about 300 Kelvin (K). This means in two months the temperature would drop to 150K, and 75K in four months. To compare, the freezing point of water is 273K. So basically it'd get too cold for us humans within just a few weeks. Some bacteria seem to be capable of surviving at extremely cold temperatures in space, so there would probably still be some limited bacterial life left on Earth. But anything else would die pretty quickly (even the rats :).
We could probably survive if we went deep underground where the Earth's internal heat is higher or if we built totally isolated habitation domes, but at the moment I don't think we're capable of something like that on any appreciable scales.