If you're trying to escape climate change making a place uninhabitable for humans, you probably don't want to aim for Mars.

I'd take living on an Earth ravaged by nuclear war and global warming over Mars any day, to be honest.

The answer to your first question is yes. Water is incredibly effective at shielding radiation - https://what-if.xkcd.com/29/

>Does that protect people from the event of a major nuclear war?

In the distant future, Mars may not necessarily afford protection either.

It's going to be sad if the doctrine of mutually-assured destruction some day applies within a multi-planetary context.

Most likely this would take the form of national territories being established on Mars by existing nuclear powers, the weapons either being imported or fabricated on-site.

Of course, the wierder scenario is one where advancements in propulsion technology allow for faster and cheaper delivery of payloads to Mars, resulting in some sort of Earth-to-Mars nuclear strike capability. Transit may remain quite latent, but it could still do the job. Imagine having weeks or months to contemplate an incoming nuclear attack.

What makes you think Mars would be more liveable than Antartica after a nuclear war or accelerated climate change?

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...