So, it finally happened. The Earth was hit by a comet, and anyone outside of a research station at the bottom of the Atlantic is now dead, or dying. So, here's what we need to know: How big would a comet have to be to wipe out life on the surface? What happened to the planet at impact? What specifically killed the surface? How long would it take for the surface to be habitable again for non-extremophile life?
Depends on what you mean.
The initial impact would have made it difficult for a while but if you are talking about how long it took for things to go back to a reasonable level for human-level species to survive, it would be something in excess of 4 million years according to the Washington Post:
This is not the end though, whilst most of of the planet was dying, some birds, crocodiles and alligators survived.
This would mean that as long as the comet (measuring some 6 miles wide) hit in the same way as it did when one wiped out the dinosaurs, there would be some slightly safer areas and some very scarce life. As long as the people you describe were able to track down one of these "safe" areas and not wipe out the fragile balance there in the pursuit of food then they could probably venture out as soon as the air was cool enough and the giant ash cloud dissipated, during a break in the massive acid rainstorms that would now be racking most of the planet.
On impact the comet would have caused a massive tsunami that would have literally brought a lot of the ocean floor up with it as well as setting off earthquakes in other parts of the world, them in turn setting off their own tidal waves. The impact would have thrown vast amounts of material into the atmosphere and superheated it. As this material came down it would have heated up again as it was coming down from some 40 miles up. When this material touched down it would have started fires in whatever vegetation was left and the overall result would be an ash cloud pretty much covering earth for about a year or so, cooling the planet to an extreme temperature and making it almost impossible for plant life.
After all of this, it gets a little difficult to determine.
Some extra information can be found here:
Likely what killed everyone was the nuclear winter that took place after the impact. The impact itself was likely not that big, maybe a crater around 6 miles wide. The thing that killed everybody, however, was the winter from the ejecta and etcetera. The amount generated from a volcano is enough to screw over lots of people for weeks, if not months. An asteroid would screw thing over exponentially.