Using results from the dielectric theory of polar solids and liquids, we calculate the mesoscopic, spatially-varying dielectric constant at points in and around a protein by combining a generalization Kirkwood-Fröhlich theory along with short all-atom molecular dynamics simulations of equilibrium protein fluctuations. The resulting dielectric permittivity tensor is found to exhibit significant heterogeneity and anisotropy in the protein interior. Around the surface of the protein it may exceed the dielectric constant of bulk water, especially near the mobile side chains of polar residues, such as K, N, Q, and E. The anisotropic character of the protein dielectric selectively modulates the attractions and repulsions between charged groups in close proximity.