The shell of bacteriophages protects the viral DNA during host-to-host transfer and serves as a high-pressure container storing energy for DNA injection into a host bacterium. Here, we probe the mechanical properties of nanometer-sized bacteriophage 29 shells by applying point forces. We show that empty shells withstand nanonewton forces while being indented up to 30% of their height. The elastic response varies across the surface, reflecting the arrangement of shell proteins. The measured Young's modulus (1.8 GPa) is comparable with that of hard plastic. We also observe fatigue and breakage of capsids after probing them repetitively. These results illustrate the mechanoprotection that viral shells provide and also suggest design principles for nanotechnology.