Relativistic Masses vs. Absolute Masses

In the classical Twin Paradox, according to the Special Theory of Relativity, when the traveling twin blasts off from the Earth, his measuring stick and other physical objects in the direction of relative motion shrink to half their lengths. Similarly, the relativistic masses are considered as increasing when traveling at a relativistic speed. But if the object is rigid, doesn't it break? And, by the way, not all masses are variable, there exist absolute masses in the universe.