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One of my favorites is the "ladder paradox" in special relativity, although I originally learned is as a pole vaulter rather than a ladder:
A pole vaulter is running carrying a pole that is 12m long at rest, holding it parallel to the ground. He is running at relativistic speed, such that lengths dilate by 50% (this would be (√3/2)c). And he runs through a barn that is 10m long that has open doors in the front and back.
Imagine standing inside barn. The pole vaulter is running so fast that the length of the pole, in your frame of reference, has contracted to 6m. So while the pole is entirely inside the barn you press a button the briefly closes the doors, so that for just a moment the pole is entirely closed inside the barn.
The question is, what does the pole vaulter see? For him, the pole has not contracted; instead the barn has. He's running with a 12m pole through what, in his frame of reference, is a 5m barn. What happens when the doors shut? How can both the doors shut?
I will admit that I have never used this thought experiment for any practical end.
For those who haven't studied relativity, this thought experiment is great at showing the "Relativity of Simultaneity".
The only way the doors can shut from the pole vaulter's reference frame is if they close at different times. The exit door opens and shuts first, before the tip of the pole has gone beyond it (otherwise it would hit the door, obviously), and then later, only once the back end of the pole has cleared the entrance door, does it close.