Here's a favorite relativity paradox. This appears in the book as Review Question 7.13.
Suppose that a train robber decides to stop a train inside tunnel. The proper length of the train is 60 m, while the proper length of the tunnel is 50 m. The train is traveling at 4/5 the speed of light. According to proper lengths, the train would not fit inside the tunnel. But the robber plans to use relativity to his advantage. The length of the moving train in the rest frame of the tunnel, and of the robber, is 36 m. The robber computes this and decides to trap the train inside the tunnel, since, in his frame, the train should fit. From the point of view of the train's engineer, however, the tunnel is only 30 m long, just half the length of the train. The engineer knows that his 60 m train will not fit completely into the tunnel. The robber thinks that the train will fit, whereas the engineer is sure it will not. But either the train will fit, or it will not; it cannot do both. Who is correct?
A version of this paradox often found in books on relativity is one where a pole-vaulter runs with a pole into a barn. The pole's rest length is greater than the rest length of the barn. Does the pole fit?