Continuity mistake: At the beginning when Indy is talking to Spalko, his hands keep alternating from being in his pockets to just resting at his sides between shots.
Factual error: During the motorcycle chase seen (which was filmed in New Haven, CT) one can see modern yellow traffic lights covered in burlap in a few of the scenes.
Factual error: In the church, you can see modern brass panic bars on the doors.
Continuity mistake: When Mutt slams his fist on the table in the diner, the ketchup and mustard bottles fall over. When the shot pans to Indiana's back, the bottles are standing up again. He then tells Mutt to sit down and the camera pans back to Mutt. The bottles are lying down again and Indiana picks them both up.
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.
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: 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.
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: After Mutt's bike knocks down the "Better red than dead" banner it falls sideways; yet when the chasing car arrives, the banner is magically up again, ready to fall horizontally on the car's windshield.
Revealing mistake: During the campus chase scene, Mutt skids his bike and turns left to drive under a stone arch. Multiple skid marks are seen on the pavement from previous shots.
Continuity mistake: The military car racing against the young kids has its right windshield covered in dust, except for the wiper's track. A frame later the glass is clean.
Continuity mistake: During the ants scene, Indy is about to grab a stick to the left of his hat. In the close-up the stick and the hat have swapped sides.
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.