We probe the local mechanical properties of microtubules at the nanometer scale by radial indentation with a scanning force microscope tip. We find a linear elastic regime that can be described by both thin-shell theory and finite element methods, in which microtubules are modeled as hollow tubes. We also find a nonlinear regime and catastrophic collapse of the microtubules under large loads. The main physics of protein shells at the nanometer scale shows simultaneously aspects of continuum elasticity in their linear response, as well as molecular graininess in their nonlinear behavior.