The current rapid growth in the use of nanosized particles is fueled in part by our increased understanding of their physical properties and ability to manipulate them, which is essential for achieving optimal functionality. Here we report detailed quantitative measurements of the mechanical response of nanosized protein shells (viral capsids) to large-scale physical deformations and compare them with theoretical descriptions from continuum elastic modeling and molecular dynamics (MD). Specifically, we used nanoindentation by atomic force microscopy to investigate the complex elastic behavior of Hepatitis B virus capsids. These capsids are hollow, approximately 30 nm in diameter, and conform to icosahedral (5-3-2) symmetry. First we show that their indentation behavior, which is symmetry-axis-dependent, cannot be reproduced by a simple model based on Föppl-von Kármán thin-shell elasticity with the fivefold vertices acting as prestressed disclinations. However, we can properly describe the measured nonlinear elastic and orientation-dependent force response with a three-dimensional, topographically detailed, finite-element model. Next, we show that coarse-grained MD simulations also yield good agreement with our nanoindentation measurements, even without any fitting of force-field parameters in the MD model. This study demonstrates that the material properties of viral nanoparticles can be correctly described by both modeling approaches. At the same time, we show that even for large deformations, it suffices to approximate the mechanical behavior of nanosized viral shells with a continuum approach, and ignore specific molecular interactions. This experimental validation of continuum elastic theory provides an example of a situation in which rules of macroscopic physics can apply to nanoscale molecular assemblies.