Visible crew/equipment: As they are running away from the ants, you can see a rig of stage lights under the jeep Col. Spalko's crew was driving.
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: 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: 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.
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: After the main characters have been thrown out of the "lost city" by the rapidly rising waters, they are completely soaked (quite natural, as they were submerged in the well). But after everything dies down, all of the characters' hair and clothes are bone dry.
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.
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).
Continuity mistake: In a close-up, Indy removes the mask from one of the mummies and is holding it about a foot away from its face. In a shot from further away, Indy is holding the mask three or four feet away from the mummy's face.
Other mistake: In the warehouse scene when they're standing around the crate containing the heavily magnetized alien, all metal objects like guns and other items are pulled towards the magnetic force field. But Indy, who is standing next to the crate, has a bag with a metal strap that stays still when it also should've been pulled towards the alien.
Other mistake: When Indy escapes on the rocket powered train, it is slowed down by a pool of water on the tracks. The only problem is that when this happens in real life, it's almost the same as slamming into a wall, and therefore Indy should have been flying several feet away from the rocket powered train. You can actually see the chair that the test persons before him used, and it has double seat belts on it.
Continuity mistake: The Russians arrive at the Nevada base and salute the guards. The one on the far right is very close to the barrier, next to where the corner of the house is; half a second later he is several meters behind and walking towards the barrier again.
Continuity mistake: In the beginning when the Russians are talking to Indy, his shirt is tucked, then in another shot it's untucked, then it's tucked again. This continues for about 3-5 shots.
Continuity mistake: The drawing that Ox makes in front of Indy differs between shots: Watch the shape of the sun.
Revealing mistake: Right after the nuclear explosion and Indy comes out of the refrigerator, the scene changes to an airfield with two cars driving across the screen. The cars are obviously CGI, as the first one turns despite the wheels still pointing straight.
Continuity mistake: Right at the start when the Russians pull Mac out of the car, they throw Indy's hat on the ground and there is a close-up shot of it where you can see Mac's feet right next to it. It then cuts to a relatively wide bird's-eye shot over the car as it is surrounded by Russian soldiers and there is nobody near the hat - Mac isn't even in the shot.
Continuity mistake: In the shot where Mutt is on the front of the car sword fighting Spalko, when the shot changes he switches to the back of the car, still fighting her.
Continuity mistake: When Indiana and Mutt are in the diner, notice the coffee cup Indy has on the table. In one shot, he has it in his hand. In another, his hand is not on it. This happens twice with the different camera angles.
Continuity mistake: After the nuclear explosion when Indiana Jones is thrown into the air inside the fridge, in one shot the doors are closed, then open, and then closed again.
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.