Factual error: Indy tells Mutt that he and Oxley were obsessed with the Mitchell-Hedges crystal skull when they were in college. The first printed reference to that particular skull was in 1936. Mitchell Hedges didn't mention it himself until 1954. Indy would've been in his mid-30's in 1936, so it wasn't possible for them to have been obsessed with it in their college days.
Factual error: In the warehouse scene, the lights and other metal objects are affected by the box's magnetic field. Yet the soldiers easily lift the box into a metal truck to load it and slide it forward. If the box's magnetic field was as strong as suggested, it would have dragged the soldiers forward as it was magnetically attracted to the truck, and it would have taken much more effort for them to slide it onto the truck bed once it was stuck there.
Factual error: The 1950s kitchen scene shows a sink faucet "Dishmaster Model 87" which wasn't manufactured until 1987. The 1950's "Dishmasters" were different.
Factual error: The movie supposedly is set in 1957, yet Mutt's motorcycle is based on a 2000 or newer Harley Davidson softail, showing the modern-day controls and Twin Cam motor, instead of the panhead motor that would have been the period-correct engine for 1957. It also has a front disc brake - you can see the master cylinder on the handlebars. Again, out-of-place on a 1957 bike.
Factual error: Indy and Mutt fly to Nazca, Peru. Nazca is only 4 hours away on bus from the capital Lima, yet the city they show on the movie is Cuzco, which is 24 hours away from Lima. Cuzco is on the east side of the country, Nazca is on the west coast right next to Lima. Also, back in 1957 only Lima had an airport, yet they show a Nazca airport that didn't exist.
Factual error: When Indy and Mutt are flying to Peru, a Pan American Aircraft is shown. It is an Antonov AN-2. Pan Am never flew this Russian aircraft! Also, the shot is backwards; the writing on the aircraft (airline name and registration) read backwards.
Factual error: During the motorcycle chase seen (which was filmed in New Haven, CT) one can see modern yellow traffic lights covered in burlap in a few of the scenes.
Factual error: The movie shows Mayan warriors talking Quechua in the Peruvian jungle instead of showing Inca Warriors.
Factual error: In the church, you can see modern brass panic bars on the doors.
Factual error: In the scene where they are flying to Peru from US, they fly over a "map" of a Central America region and the country of Belize is shown. That country was the British Honduras until 1973. It was then renamed to Belize.
Factual error: When Indy is getting decontaminated after the atomic explosion, the workers are using the M17 protective masks. These masks were not in use until the late 1960's.
Factual error: During the fight in the rocket engine testing facility, the countdown timer has electronic digital numerals. In 1957, the countdown timer would have had flipping numbers, which are mechanically changed as the timer counts down, or a nixie tube display. See films like Fail Safe and Goldfinger for examples of such clocks or timers. LED's weren't even invented until 1962.
Factual error: When Mutt throws the snake to Indiana Jones in the sand pit, it's an olive python (which he misidentifies as a rat snake). But olive pythons are not native to South America - they're Australian.
Factual error: In a shot of the exterior atomic test homes, we see child dummies playing on a Slip-n-Slide. That toy was not introduced by Wham-O until 1961.
Factual error: The Russian soldier would not be able to scream with ants running into his mouth - he would be choking.
Factual error: In the wedding scene, the minister uses the modern, politically correct "I now pronounce you husband and wife" instead of the traditional "man and wife" - unheard of in 1957. (01:54:50)
Factual error: In the warehouse when Indy and the Soviets open the crate with the alien's body, all the overhead lights are pulled towards it. If the magnetic field was that strong, the lights also would have been pulled towards the box before it was opened. (Put a toy car on top of a table and pull it with a magnet from under the table to see the field in action).
Factual error: At the beginning of the movie where the convoy stops at the gate to the military base, the fence has "Razor Wire" on the top, not the period-correct barbed wire. Razor wire came into use in the late sixties by some industries, but the military did not use it until the eighties.
Factual error: After Indy finds the skull, the map to show his travel has all the words written in Spanish, with the accents properly placed, except for the words 'Cuzco' and 'Río Amazonas', misspelled in English, 'Cusco' and 'Amazon'.
Factual error: Skara Brae is on Mainland Orkney not the West Coast of Scotland as Indy states in class just before he's let go from the university.
Answer: People felt it was ridiculous and cartoonish, even by Indiana Jones standards. Even if it was possible to survive a nuclear blast via the lead lining of a fridge (it's not), or that the fridge would simply be thrown away rather than be melted/torn apart like everything else in the vicinity (it wouldn't), the impact of being flung what appears to be a mile or so through the air, then violently crashing into and rolling over the ground, would certainly kill anything inside. The controversy arose because usually, in "classic" Indy films, the fantastical elements were exactly that: fantastic, magical, and/or supernatural. This was presented as taking place in our reality, with no "power of God" or magic spells, and for many, that was just too much disbelief to suspend.