In the event of a major nuclear war, living in Antarctica won't protect you from environmental radiation, which will be at highly elevated levels for decades throughout the entire planet.

And how do you protect yourself from radiation on Mars, given its thin atmosphere and lack of magnetic field?

Mars's thin atmosphere does provide significant levels of radiation shielding, such that surface radiation levels are comparable to those in low-Earth orbit:

http://www.space.com/18504-mars-rover-curiosity-astronaut-ra...

The Moon, on the other hand, is baldly exposed:

https://science.nasa.gov/science-news/science-at-nasa/2005/0...