Question: It's 2016, but for some reason, the Cold War never ended. You've been inspired by your (for some inexplicable reason) favorite movie, Indiana Jones 4, to start a new business: building refrigerators. However, the market is so saturated that in order to distinguish yourself with a nice marketing campaign, you decide to make them **NUKEPROOF!** 

The refrigerator should be:

1. Usable as a real refrigerator would be.
2. No bigger than standard double door refrigerator, so no room-sized walk-in refrigerator. Though, you *can* add some additional size for additional armor, for example, in **reasonable** margins, so nothing like 10 meters of iron from one side.
3. Protect one person from up to 1.2 megatonnes of a TNT nuke at a minimum distance of 2 km from the explosion.

How are you going to design it and what materials are you going to use?

Answer: 
Well, I'm going to say, my refrigerator is a SpaceX dragon launch abort system with a baggy of ice inside.

The total volume of the capsule is 25 cubic meters, which is the equivalent of a 2.9m x 2.9m x 3.0m cube, or a refrigerated room, as seen in grocery stores.

According to the spaceX website, the capsule can move a crew (of 3) vertically upwards about 5000 feet (0.9 miles, or 1.5km), which means that it could travel about 7070 feet (2.15km) laterally. This would carry you out from 2km off the blast range to 4.15km. While most buildings would still be completely demolished, you, being in the air, would be perfectly fine with no debris to fall on you. Maybe a bit toasty, but hey, why do you think it has a bag of Ice?
GrinningX has quite a good answer but I would like to amend it slightly. The stats quoted appear to refer to a ground burst. Ground bursts are not thought to be targeted against cities because air bursts are more damaging to them. Ground bursts are expected to be used against hardened targets like ICBM silos. 

However, if a 1.2MT ground burst did occur in your city it the fridge owner would die for sure. Ground bursts generate enormous quantities of fallout. (Air bursts generate almost no fallout.) Fallout is created when energetic neutrons in the fireball touch heavy matter, converting it into unstable isotopes. Air burst neutrons interact with gasses, and the radioactive isotopes they make from this gas tends to stay aloft. 

The fallout from such a ground burst would be lethal for weeks to months - far too long to be hiding inside a refrigerator. 

On the other hand, an air burst produces almost no radiation on the ground (neither prompt nor fallout) so the radiation shielding would be largely unneeded. The most important need would be (in this following order of events) 1) protection from the flash (just being inside your house and away from windows is probably good enough), 2) protection from blast (sturdy steel construction and being securely bolted to a sturdy foundation would keep you alive), 3) protection from the heat and fumes from your house burning down, and 4) a mechanism to allow you to exit the fridge even though it has been buried by the debris of your house. Numbers 3 and 4 are quite a bit more difficult and complicated than 1 and 2. 

For surviving the fire, maybe some kind of hand-cranked air pump which pulled air in from outside and percolated it through a container of water (to cool the air and remove smoke)? That would help, but would not do anything to reduce carbon monoxide or lethal carbon dioxide levels. Heat would be less of a problem. Fridges are designed to be insulated, so just make sure the gaskets are fire resistant and increase the R value enough to survive. 

I'm honestly not sure how to handle egress. It's hard to guess what debris will be on the fridge and how it will be shaped. I don't have any good ideas for this one.