We present an investigation of the spectral and spatial structure of the X-ray emission from 3C 58 based on a 350 ks observation with the Chandra X-ray Observatory. This deep image, obtained as part of the Chandra Large Project program, reveals new information on nearly all spatial scales in the pulsar wind nebula (PWN). On the smallest scales, we derive an improved limit of T < 1.02 × 10⁶ K for blackbody emission from the entire surface of the central neutron star (NS), confirming the need for rapid, nonstandard cooling in the stellar interior. Furthermore, we show that the data are consistent with emission from a light-element atmosphere with a similar temperature. Surrounding the NS, a toroidal structure with a jet is resolved, consistent with earlier measurements and indicative of an east-west orientation for the projected rotation axis of the pulsar. A complex of looplike X-ray filaments fills the nebula interior and corresponds well with structures seen in the radio band. Several of the structures coincide with optical filaments as well. The emission from the interior of the PWN, including the pulsar, jet, and filaments, is primarily nonthermal in nature. The power-law index steepens with radius, but appears to also show small azimuthal variations. The outermost regions of the nebula require a thermal emission component, confirming the presence of an ejecta-rich swept up shell.