Factual error: In the warehouse scene, the lights and other metal objects are affected by the box's magnetic field. Yet the soldiers easily lift the box into a metal truck to load it and slide it forward. If the box's magnetic field was as strong as suggested, it would have dragged the soldiers forward as it was magnetically attracted to the truck, and it would have taken much more effort for them to slide it onto the truck bed once it was stuck there.
Factual error: The movie supposedly is set in 1957, yet Mutt's motorcycle is based on a 2000 or newer Harley Davidson softail, showing the modern-day controls and Twin Cam motor, instead of the panhead motor that would have been the period-correct engine for 1957. It also has a front disc brake - you can see the master cylinder on the handlebars. Again, out-of-place on a 1957 bike.
Factual error: Indy and Mutt fly to Nazca, Peru. Nazca is only 4 hours away on bus from the capital Lima, yet the city they show on the movie is Cuzco, which is 24 hours away from Lima. Cuzco is on the east side of the country, Nazca is on the west coast right next to Lima. Also, back in 1957 only Lima had an airport, yet they show a Nazca airport that didn't exist.
Continuity mistake: When Indiana and Mutt are in the diner, at the start of the scene there is a pretty blonde girl sitting at the counter just past the corner. A moment later, her seat and the one next to it are occupied by the two Russian guys in suits, and she is talking to a guy with glasses who was earlier by himself. Not enough time had passed for that to change.
Plot hole: Hiding in a fridge (or anything else) in order to be conveniently blown out of the way by an exploding nuclear device is absurd beyond belief. The fridge is just so much extra reaction mass and would be vaporised by the expanding nuclear explosion - it wouldn't be daintily picked up and thrown a few kilometers to safety. If it was, why doesn't it land in a shower of similar artifacts which have also been dislodged and thrown around? Incidentally, even if it was thrown out of the way as shown, anyone inside it would be turned into a smear of strawberry jam by the acceleration required to beat the shock and heat wave of a nuclear blast, and then liquefied by the deceleration involved in hitting the ground at that speed.
Continuity mistake: When Indiana Jones is locked in the chair at the Russian camp, there is a reel-to-reel tape recorder machine running to Indy's left. When you first see it, the amount of tape on the left reel is low. Then, when Irina Spalko walks in, she goes over to it and flicks a switch. As the camera moves out, the tape on the left-hand reel is suddenly nearly equal to the one on the right.