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.
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.
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.
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.
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.
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.
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.
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: 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: During the wedding scene, we see Mutt taking photos with a 1950s Kodak Brownie box camera. When wind blows Indy's fedora off the hat rack and up the aisle, Mutt stoops to pick up the hat, with the camera no longer in his hands. Indy snatches the hat away from Mutt as he passes, and the shot then cuts to a full view of all the wedding guests leaving the church, with Mutt in the center of the shot, still standing in the aisle the whole time. He takes out a comb and passes it through his hair, then picks up his jacket from the pew before he leaves, but the boxy Brownie camera is still nowhere to be seen. It's not in his pockets, and it's not in his jacket (the angular bulge would be more than noticeable).
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.
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.