This is actually a pretty great thought experiment. At first it might seem kind of pointless to talk about what would happen if the sun vanished, but it doesn't actually result in the immediate destruction of everything. Which is weird.
Vsauce walks through a pretty nuanced description of how earth's natural systems would slowly fail, but over weeks and even years, not seconds. The cold would get us in the end, but extremophiles that live in deep sea volcanoes and thermal vents could survive for billions of years. If you're not heliocentric and human-centric things don't look so bleak. [Vsauce]