One of the first crises of this sort was when the Pythagoreans realized that the the square root of 2 is irrational. It was so earthshaking that this finding was kept hidden and secret for many years. Today, irrational numbers are not a bother.
Another "impossible" situation was the square root of -1. Today we have an interpretation of that, and all electrical engineers are quite comfortable using the square root of -1, as part of an "imaginary number". In reality these numbers are not "imaginary" at all, but offer a way of describing the reality of electrical circuits.
I'm sure that someday the "string theorists" will find a plausible interpretation of their results.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it ...
An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out, and that the growing generation is familiarized with the ideas from the beginning: another instance of the fact that the future lies with the youth.
— Max Planck, Scientific autobiography, 1950, p. 33, 97