Here's the thought experiment that sold me on how weird this stuff is.
Imagine explorers on Mars find the ruins of an ancient alien civilization. In those ruins they find several small devices that have three buttons. Beside each button are two colored lights. red and blue. Above the buttons is a display. The linguistics team figured out enough alien writing to tell that the buttons are labeled with the alien's equivalent of A, B, and C, and that the display is a numerical display that goes from 0 to 38413 displayed in base 14 (which fits with other evidence found that the aliens have two hands with 7 fingers).
There is also some kind of docking station, which can hold two of the devices, and has a single button.
If two of the devices are placed in the docking station and its button is pressed, all the lights briefly flash on the devices, and the counter resets to 0. The lights stay on until the device is removed from the dock. Nothing happens if only one device is placed in the dock.
To try to figure out what these devices do, pairs are placed in the dock, reset, and then given to a couple people who go off and press the device buttons are record what happens.
Here is what those people observe.
1. If they press one of the buttons (A, B, or C), exactly one of the two lights next to that button comes on. When the button is released, the light goes out, and the counter goes up by 1, until it reaches 38413. After the next press/release, the counter goes blank and the device is unresponsive until reset again in the dock.
2. As far as anyone can tell, there is no pattern to which light lights. It acts as if pressing a button consults a perfect true unbiased uniformly distributed random bit generator to decide between red and blue.
3. When they compare their results with those of the person who had the box that was their box's dock mate for reset, they find that if on each person's n'th press
-- if they both pressed A, or both pressed B, or both pressed C, they got the same color light.
-- if one of them pressed B, and the other pressed either A or C, they got the same color light 85.36% of the time.
-- if one of them pressed A and the other pressed C, they got the same color light 50% of the time.
4. These results do not depend on the timing between the two people's presses. Those correlations are the same if the people happen to make their n'th press at the same time, or at wildly different times. Even if one person goes through all their presses before the other even starts, their n'th presses exhibit the above correlations.
5. These results do not depend on the distance between the boxes. If a box pair is split up, with one person taking theirs back to Earth while the other remains on Mars, and the two then run through all their presses at nearly the same time, completing quickly enough that there can be no communication between the two boxes during the run due to speed of light limits, they still exhibit the correlations.
Challenge: try to figure out how such boxes could be built without using quantum entanglement. Assume the aliens have nearly unlimited storage technology, so you can include ridiculously large tables if you want, so you can even propose solutions that involve the dock
preloading the responses for every possible sequence of presses (all 3^38414 of them). Anything goes as long as it produces the right correlations, and does not involve quantum entanglement.