1. There is such a thing as a lead-lined refrigerator.
2. This was not a feature that ordinary household refrigerators had in the 1950s.
3. Even a refrigerator made entirely of lead would probably not save you receiving a lethal radiation dose within the radius of the blast depicted in the film.
4. When attempting to survive a nuclear blast, do NOT hide in a refrigerator.
The lead-lined refrigerators are made to store materials used in nuclear medicine. The shielding they offer is pretty limited, only .125" of lead. Also, they're made to fit under a lab bench, and I doubt that any normal adult could shoehorn themselves inside of it. For comparison, let's explore the kind of radiation protection offered by a hypothetical refrigerator made entirely of lead.
According to Cresson Kearny, 1 cm (.4") of lead reduces the intensity of gamma rays by 50%. Therefore, 2" of lead would reduce gamma ray intensity to 6.25% of the original intensity. Note that this does not include the effects of neutron, beta, and alpha radiation. Calculating this is a LOT more complicated. As the 1977 edition of The Effects of Nuclear Weapons explained:
Neutron shielding is a different, and more difficult, problem than shielding against gamma rays. As far as the latter are concerned, it is merely a matter of interposing a sufficient mass of radiation between the source of gamma radiations and the recipient. Heavy metals, such as iron and lead, make good gamma-ray shields because of their high density. These elements alone, however, are not quite as satisfactory for neutron shielding.
Using the graphs provided in The Effects of Nuclear Weapons, it appears that from a distance of 1000 yards, a 20 kT fission explosion would result in a dose of about 4,000 rads from gamma and 4,000-6,000 from neutrons. Therefore, the shielding offered by our hypothetical lead fridge would reduce exposure to its occupant to 260 rads, with a presumably greater dose from neutrons. More than half of individuals exposed to 500+ rems will die of acute radiation poisoning, with several months of convalescence minimum. In short, Indy would not be up for a trip in the near-term to battle Soviet agents in exotic South American locales. Never mind the massive injuries he would have sustained from being flung around inside a heavy metal box.