You can actually give a semi-plausible justification of special relativity based on what was known in 1901. Maxwell's equations are fundamentally incompatible with what I might call "Newtonian relativity." They define a fixed speed for light, which is impossible in Newtonian relativity for observers in different inertial reference frames. Magnetism is also a puzzle, as the magnetism depends on the relative velocity in a way that makes it appear to create different forces in different inertial frames. Without length contraction from special relativity, magnetism has uncomfortable implications.
Suppose I told you that I knew for a fact that the following statements were true:
You’d think I was crazy, right?
Now suppose it were the year 1901, and you had to choose between believing those statements I have just offered, and believing statements like the following:
Based on a comment of Robin Hanson’s: “I wonder if one could describe in enough detail a fictional story of an alternative reality, a reality that our ancestors could not distinguish from the truth, in order to make it very clear how surprising the truth turned out to be.”1
1Source: http://lesswrong.com/lw/j0/making_history_available/ewg.