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: 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: 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: When the civilian car escapes the oncoming car, the three trucks of the military convoy are missing. Only the three small vehicles are on the street. (00:01:25)
Continuity mistake: When the race begins between the staff car and the hot rod, they immediately pull away from the convoy, as is logical. However, in the next shot the convoy is right behind the staff car again - which is still accelerating racing the hot rod. When the convoy pulls off the main road towards Area 51, the convoy again is right behind the staff car. (00:01:40)
Continuity mistake: While the staff car and the black car are racing, there's a wide shot of the place, with only the road and some mountains in the background. In a few shots later, the Atomic Cafe has appeared from nowhere. (00:01:52 - 00:02:20)
Continuity mistake: The Russian convoy consists of one staff car, two jeeps, and three trucks. When the convoy arrives at Hangar 51, a second staff car has appeared, from which Irina Spalko steps out. (00:02:44 - 00:05:50)
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: When Indy asks for shotgun shells, behind him are, from left to right, Ivan, Spalko, Dovchenko, and Mac. Just behind Spalko, stands Roosevelt. But, when Indy asks for pliers, Roosevelt has appeared next to Indy, Spalko has moved to the right side of the group, and Ivan has disappeared. (00:09:57)
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: While the soldiers are loading the coffin inside the jeep, you can see Spalko on the left side of the jeep, handing his rapier to Dovchenko. In the next shot, she is next to the back of the jeep with the rapier again and repeats the handing action. (00:11:39)
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)
Continuity mistake: Dovchenko smiles and raises his pistol. In the next shot, he repeats the action. (00:13:34)
Continuity mistake: When Indy and Dovchenko are shot out of Hangar 51, the expression on the Russian's face changes between shots. (00:17:00)
Continuity mistake: As the Dean enters, several pictures hang on the wall. When they leave the pictures have been changed. (00:24:40 - 00:25:45)
Continuity mistake: In the university classroom scene, Indiana turns to the dean and says, "Yes?" He is holding the chalk with both hands and his elbows are bent. In the next shot, both arms are at his side. This is seen on the DVD. (00:25:00)
Continuity mistake: After Indy shouts "What?", he and Stanforth walk through the corridor and pass a painting of Brody that wasn't there when Stanforth was walking alone before entering Indy's classroom. (00:25:45)
Continuity mistake: In Nazca, the two locals Indy talks to about Oxley teleport next to the bus door between shots and the large wagon behind Indy disappears. (00:39:28)
Continuity mistake: As Indy's truck is getting closer, the camera focuses on Spalko. Behind her are the duck with Oxley and Mac inside and a damaged truck. But in the next shot, when Oxley says "Henry Jones Jr.", the truck is gone. (01:16:17)
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)
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.