Audio problem: When Irina and Mutt are sword fighting in the jungle, the blade strikes all make echoing sounds.
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.
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.
Continuity mistake: Before Indy finds the crystal skull, the mummy's head has a piece of cloth on its forehead. In a wide shot immediately afterwards, this cloth disappears.
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.
Continuity mistake: During the chase above the waterfall, the assault boat has its machine gun snapped off. When the assault boat goes over the waterfall the gun is visible again for a few seconds. The gun is gone when they get to the bottom.
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: Indy's mostly buried in the quick sands and he is handed a snake. Half a second later, when he has just managed to grab it he isn't buried as deeply.
Continuity mistake: During the scene where Indy and Mutt first find the skull and they are having a conversation about it. Indy is holding the skull with the face pointing to his right. As the camera changes back it is facing to his left, then it changes back again.
Continuity mistake: Indy's head position and hairstyle keeps changing in the shots after the test jet comes to a stop.
Continuity mistake: When Spalko climbs a hanging vine to escape the large, marauding ants, one jumps on her leg. As it crawls up, she smashes the squishy bug between her knees (you can hear it splatting), but there's no stain or body parts left on her very clean pants.
Continuity mistake: In the beginning when Indy's hat is thrown from the trunk onto the ground, it faces the 9 o'clock postion. But when Indy reaches to pick it up half-a-second later, it faces the 12 o'clock direction.
Continuity mistake: When Indy is held captive in the Soviet's tent, Mac enters and they have a conversation. Indy then turns his head to his left, looking straight at Mac. Just before Indy makes his threat about what he'll do to Mac when he gets loose, there is an angle change, and Indy is suddenly staring straight ahead, with no time to turn his head in between shots. He then turns his head back to look Mac in the face again.
Continuity mistake: Throughout the whole movie, Irina's hair seems to be kept absolutely pristine, even when she is flung from cars. This is very noticible at the car chase in the military base when she is thrown from the car by Indy. One minute, a few strands of hair are slightly messed, then they are fixed again. It is likely, however, given the many other nods to films of that period, that the "concrete hair" effect is a deliberate decision.
Continuity mistake: In the diner scene between "Indy" and "Mutt," Indy's shirt-collar points are smooth under his jacket lapel in some close-up shots, but in others one collar point is curled up.
Continuity mistake: When Mutt first shows up, he is riding his bike between two train wagons, or between a train and a railing next to the station. This changes back and forth with no logic at all.
Other mistake: In the scene when they are all inside the temple with the skeletons, the Russian woman takes the skull and walks up the stairs directly in front of the creature. Yet there are no direct steps. All steps leading up to the skeletons are a little off to the left and right, and there are two sets of them.
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'.
Continuity mistake: During the jungle chase, while Spalko fires at Indy's jeep, her hair remains perfectly still in the angle shot from behind, but moves wildly in the front shots. This swaps back and forth.
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.
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.