Continuity mistake: During the siafu ant attack, while fisticuffs ensue between Indiana and Dovchenko, note the direction of Dovchenko's belt, with its buckle prongs to his left. Then, when Dovchenko falls backwards and the ants swarm all over him, his belt is worn around his waist the opposite way, with its prongs to his right (neither are flipped shots). (01:26:55)
Continuity mistake: When Indiana is with Mutt while the boy rides the motorcycle, Indiana's shoes change; note the soles change from heels to flat rubber-type soles, then back to heeled soles.
Continuity mistake: When Indy is about to grab the branch to hit the soldier, his hat keeps moving around between shots.
Continuity mistake: When Mutt meets Indy, Mutt has a train behind, yet the reflection of the glasses from Indy's wagon don't reflect it, instead showing a sky and trees.
Continuity mistake: In Akator, Indy, surrounded by Marion and Mac, tells Mutt to find a stone. When Mutt comes back, in less than a second, Marion and Mac are nowhere to be seen.
Other mistake: Spalko is hanging from a tree to avoid the sting of the ants. When a CGI ant is on her knee she closes both legs and smashes it. If you watch closely you'll notice that when she closed her legs the ant was already over her thigh, so if it had been a real ant it would have been impossible to kill that way.
Continuity mistake: After Mutt falls off the graveyard's ladder and is attacked by a native, the scattered broken parts of the ladder disappear to allow for the fight scene to take place.
Continuity mistake: In the Nevada warehouse, while Spalko opens the bag a strap next to it appears or disappears depending on the angle.
Continuity mistake: When Indy and Mac are aiming at the soldiers in Nevada the distance to the Jeep behind increases noticeably between shots.
Continuity mistake: While Indy and Mutt are walking by the market stalls, there's no continuity with the people behind them. Sometimes even a group of men disappear only to reappear several shots later.
Continuity mistake: When Jones is strapped to the chair in the tent Irina uncovers an alien to shows him. She then covers the alien with the cloth, but when the camera moves to a wide shot the soldier at the other side of the table does the same thing.
Continuity mistake: At the diner, Mutt places the photo on his hat, yet it swaps from standing out to tucked inside, between shots.
Continuity mistake: During the fight in the bar, when Indy and Mutt exit, a man to the right hugs a woman in green to protect her. A frame later they're 3 meters apart.
Continuity mistake: When Ox finds out about the stone that pours out sand, Mutt is up the stairs. Then the angle changes and he is down, starting to walk upwards.
Continuity mistake: While Dovchenko is approaching Hangar 51's gate, there is a guard standing behind the STOP white sign written on the barrier. In the next shot, that guard has teleported next to the fuse box. (00:03:03)
Continuity mistake: Indy asks for shotgun shells and a soldier drops some inside Dovchenko's cap (which is in Indy's left hand). In the next shot, they are suddenly in the same hand that held the cap, which is now in Indy's right hand. (00:10:00)
Continuity mistake: Before being knocked by Mac, a Russian soldier throws the Skull to Spalko, who has her rapier in her right hand. In the next shot, the rapier is in its sheath. (01:18:48)
Continuity mistake: At the start, when the carful of teens pulls up alongside the convoy's lead vehicle, in the first shot of that lead driver (his reflection is in the round wing mirror), the side of the hood reads "U.S. Army" and "276***," the partial serial number beneath. Then, in the next shot and subsequent shots, the lead vehicle's serial number reads "1B7731" to the side of the hood. Just to add, when Spalko arrives at Hangar 51, her vehicle has that "276" serial number with license plate W-539. (00:01:05)
Continuity mistake: At the start, just as the carful of teenagers joins the convoy of military vehicles on the road, there are very long shadows being cast on the road, which vanish in the following shots. Their long shadows return when the convoy turns onto another road, but disappear again as they arrive at the main gate. (00:00:45)
Continuity mistake: At Hangar 51, after the Soviets open the crate with the mummified alien, Indiana grabs his whip and secures guns for Mac and himself. In the close-up and wide shot, Indiana is gripping the whip within his palm, but in the next shot, the whip is held with only his index finger; then it's back to being gripped in his palm. Also, while Indiana is aiming the gun at Spalko, the military jeep behind him changes its distance, depending on the shot. (00:13:15)
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.