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: 
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.
The it's not possible answers are correct if you interpret the requirements strictly in light of the scene from the movie, where the fridge is directly exposed to the nuclear explosion. Having said that, your company could legitimately market a fridge that would be of benefit, within certain limitations.

1. The fridge would require to be installed in a cellar, or ideally in a reinforced shelter dug into your garden. At a time of heightened nuclear tension digging a shelter may be something people are prepared to do, similarly to how Anderson shelters were dug in many gardens in the UK in World War 2. A shelter intended for protection from a nuclear device should ideally have a zigzag tunnel leading to it: the earth will provides a significant amount of protection from direct gamma, neutron and thermal radiation.
2. The fridge should have a large dedicated compartment for fresh water, sufficient for an entire family to drink for at least a couple of days while you trek out of the fallout zone.
3. The fridge should have a (non-refrigerated) compartment for other emergency supplies, particularly enough P-3 particulate masks for the entire family. As above, these will be used during the first couple of days after the explosion while you trek out of the fallout zone, to prevent internal contamination. Other useful items: ear plugs (your eardrums will likely rupture thus providing a route for contamination particles into the lungs / stomach, so you will want to block your ears), disposable razors (contamination will be trapped quite effectively in hair so you will need to ensure everybody gets a close all-over shave, to reduce contamination), basic medical kit including splints and burns dressings.
4. The fridge should have a lead-acid battery backup, for when the power goes down, and beefy EMP protection.
5. The fridge should feature a pull-out strong metal frame so that it can serve the function of a Morrison shelter and protect the users from falling rubble if the building above collapses on the cellar, or the roof of the garden shelter falls in. It should also have a shovel and combination hammer / crowbar / wrench in case you need to dig your way out of the rubble.

It doesn't really work in the same way as the Indiana Jones fridge, but I'd still be pretty happy to have such a fridge in a shelter in my garden if I was worried about possible nuclear attack. Obviously nothing would protect you from a direct hit, but the fridge would considerably enhance a shelter capable of supporting survival in the scenario you outlined as long as sufficient warning was provided to allow you to get into your shelter before the blast.