Visible crew/equipment: When Indy and Sallah are deciphering the head piece there is a shot over Indy's shoulder. As he turns you can see the reflection of a fluorescent light in his glasses. (00:48:40)
Continuity mistake: Marion's bar in Nepal is built with thick stone walls, but after the fire, all you see is the burned-out shell of a wood-frame building.
Factual error: Indy's satchel is a Mk VII gas mask bag, which was created for use in World War II, so it didn't even exist at the time the film takes place (1936). The similar Mk V existed at the time, though.
Revealing mistake: Where is all the light coming from in the Well of Souls, even after the entrance was re-sealed, and the only torch still burning went out?
Factual error: When flying from Nepal to Egypt Indy passes over Arabia. The state of Jordan is shown, but in 1935 Jordan was called The Emirate of Transjordan.
Factual error: It should be a tricky task to put an object the size of the Ark on board a German VIIC submarine; perhaps you can find enough place inside but how do you manage to introduce that bulky case through the narrow hatches of the sub? Even the torpedo load hatches were just a little more than 21 inches in diameter.
Plot hole: Indy stows away on the Nazi sub on the way to the secret Nazi base somewhere near Greece. Where did he stay all that time? Inside? On a VII-Class boat, there's barely enough room for the regular crew, and hardly any places to hide. Also, everybody knows everybody, so no passing yourself off as "the new guy." Outside then maybe? Well, the boat did dive, it's in the dialog, and while on the surface, German U-boats always had a bridge watch on the tower, so no luck there either. Note that by all accounts I could find, German submarines were required to always post a bridge watch, if only because otherwise they would be running essentially blind. It's not like the helmsman's station in the tower had any windows.
Continuity mistake: When Indy and the Indian are entering the cave and clearing the cobwebs, you can see at least one spider on Indy's back. The next shot shows a clear, long shot of Indy's back with light reflecting on his jacket and no spiders. The next shot, the Indian notices the spiders. (00:04:20)
Continuity mistake: At the beginning of the film, Indiana Jones uses his whip to disarm a man with a handgun. The scene showing this action includes several quick shots, but as a result of the use of these multiple shots it can be seen that the whip descends twice. (Slow motion helps, but isn't required.)
Factual error: At the beginning of the movie, when Indiana is looking for the temple, you can hear a bird call. It also can be heard when Indy is running from the natives. The bird is a willow ptarmigan. It lives in very cold areas in the far north of North America. Even a very directionally-challenged ptarmigan could not possibly end up in South America. You can also hear the unmistakable laughing sounds of the kookaburras, only found in Australia.
Continuity mistake: In the beginning, Indy's partner takes the idol, throws down the whip, and leaves him hanging on a ledge with the ceiling closing down to the ground. While he is hanging the you can see that the ceiling barely has enough room for Indy to fit. Indy struggles to get up which during this time he would have no room to get under, yet when he finally gets up the ceiling has risen giving him plenty of room to get under. (00:08:45)
Continuity mistake: In the scene where Indy is telling about the Staff of Ra, the background color of the blackboard changes between shots. It looks like there are some chalk residues (after the blackboard has been washed) in one of the shots, but not in the other.
Continuity mistake: The stone block Indy pushes out to create an opening which allows him and Marion to escape the Well Of Souls apparently rolls down the hill outside. We see it, in a following scene, laying about 15 feet away from the Well. But when Indy is fighting the big German guard near the airplane, one scene clearly shows the Well in the background and the dislodged stone is now laying on the hill only a few feet from the opening. (01:15:45 - 01:18:10)
Continuity mistake: When the government comes to see Indy at the University he is teaching, they enter the large room, I think it's a gymnasium. He puts the huge book on the table and one of the buckles comes undone. But when he goes to open the book to show the g-men a picture of the Ark he undoes both buckles. (00:15:30 - 00:19:30)
Continuity mistake: When Indy is being lowered into the map room, the exterior shot of the opening does not match the interior. From the outside, the wall around the opening is a lot thicker. The angle is also different. (00:50:55)
Continuity mistake: When Indy and the German driver crash through the villager's construction ladders the truck is missing the Mercedes-Benz hood ornament that appears in every other shot.
Continuity mistake: When Indy is trying to stop the truck that is supposedly carrying Marion in a basket, Indy shoots the driver as the truck heads towards a "dead end". The driver's foot hits the accelerator and the next shot opens up to a big open space where the truck tips over and explodes. The "dead end" has disappeared. (00:40:10)
Audio problem: When the monkey is crawling on Marion's head at Salah's place, the sound does not match Marion's mouth.
Continuity mistake: Sallah and Indy are peering into the well of souls. Sallah is to the right of Indy. Indy then drops the torch into the well, and when the camera comes back up, Sallah is on the left of Indy. (01:00:00 - 01:00:45)
Revealing mistake: Just before Belloq says "beautiful", look carefully at the woman's ghost in the center of the screen. You can see one of the wires on her back that makes the stunt woman fly, a black line above her left shoulder.
Answer: Indy and Marcus Brody believed that the Ark needed to be studied. They certainly didn't want to put it back where it was found. They believed that the U.S. government would find the best archaeologists, researchers, and scientists in the world to study the Ark. They are both upset that instead, the government has decided to simply lock the Ark away. This is why Indy says, "Fools. They don't know what they've got there," as he is leaving the building.
BaconIsMyBFF