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.
Continuity mistake: In the jungle chase scene, Indy and the others blow up the front-running plough machine. They then continue to drive around in a ploughed jungle alley. They should be hitting a lot of trees.
Continuity mistake: Just before the Russian backhands Indy in front of the warehouse at the beginning of the film, his hands are crossed behind his own back, but immediately in the next shot, they're crossed in front of him.
Continuity mistake: After visiting Ox's jail, Indy arrives at the ruins and it's pitch dark. He enters the burial room several minutes later and sun beams are now coming through the ceiling.
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.
Continuity mistake: When Mutt slams his fist on the table in the diner, the ketchup and mustard bottles fall over. When the shot pans to Indiana's back, the bottles are standing up again. He then tells Mutt to sit down and the camera pans back to Mutt. The bottles are lying down again and Indiana picks them both up.
Continuity mistake: When the gang arrive at Akator, and Indy pulls the first rock from the wall and sand starts pouring out, Mutt runs out from that shot to find himself a rock but in the next shot, when Indy says to Mac, "Go get your own!(rock)", Mutt is suddenly back in the shot and runs off to find a rock, again.
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.
Continuity mistake: In the opening scene with the drag race, the lead car in the caravan speeds ahead in response to the challenge. In the next scene, the caravan turns right and the challenger continues straight. At this point, the entire caravan is together despite the fact that the lead car would have been far ahead by that point.
Continuity mistake: When they fall into the water, while the stairs disappear, the skull splashes twice.
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.
Other mistake: When Mutt is riding his motorcycle the first time he meets Indy at the train station, his hand is pulling in the clutch as he rides. But with the sound and speed of the bike, the clutch would have to be out.
Continuity mistake: When the civilian car is chasing the military convoy approaching the Nevada test site it crosses the convoy between the third and the second truck. It then passes the next two trucks on the left. In the next shot it is again behind the first truck. (00:00:55)
Continuity mistake: After the duel between Mutt and Irina Spalko, Mutt falls down into the car with Indy and yells, "She's getting away!" as she is seen driving off. However, in the following shot when they are driving near the edge, Irina comes from behind them.
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)
Continuity mistake: When Mutt and the Russian woman are fighting in the normal view, you see the the AK-47, but when it changes to overhead, it disappears.
Continuity mistake: When Indy and Mutt meet in the diner, the spoon on the table changes its position between shots.
Continuity mistake: In the scene where the two cars collide frontally in the warehouse, Mac is shown inside the car and immediately afterwards he is shown standing outside the car, while Indy climbs the ceiling woodwork.
Continuity mistake: When Spalko is talking to Indy at the start, there are four Russian soldiers behind her who have their guns pointed at Indy. When Spalko says, "order of Lenin," and walks a few steps away from Indy, the soldiers raise their guns to let her pass. It then cuts to a shot of Spalko over Indy's shoulder and their guns are pointed at him again. It then cuts back to a profile shot of the pair and the guards once again have their guns raised.
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.