We present results from a 49.4 ks Chandra/ACIS-S observation of the nearby ($z$=0.059) X-shaped FRII radio galaxy 3C~403. This is the first Chandra observation of an X-shaped radio galaxy, and one of the goals of this pioneering study is to determine the relationship between the X-ray emitting gas and the X-shaped radio morphology. The X-ray isophotes of the hot gas within $\sim3.5''$ of the central galaxy are highly elliptical (eccentricity$\sim$0.57) and co-aligned with the elliptical optical isophotes. This supports the hypothesis that X-shaped radio sources are created by propagation of jets through asymmetric density distributions. Within large uncertainties, there is no evidence that the lobes or wings are overpressurized relative to the ISM. We have detected X-ray emission from several of the radio knots to the E of the active nucleus, and diffuse emission from the radio lobe to the W. The X-ray emission from the eastern knots cannot be explained by an inverse Compton model unless they are far from equipartition. Using archival HST data, optical emission is detected from two knots, and the radio/optical/X-ray spectra are well fitted by simple synchrotron models. This is one of the strongest examples to date of X-ray synchrotron emission from multiple knots in the jet of an FR II radio galaxy. X-ray emission is also detected from the radio wings at a flux consistent with inverse Compton scattering of CMB photons from relativistic electrons if the wings are near equipartition. The nuclear spectrum is well described by a multi-component model that includes a heavily absorbed power law and a bright, broadened Fe line. A second, less absorbed, power-law component, likely to represent unresolved emission from a pc-scale jet, is also required.

35 pages, 12 figures, 3 tables - accepted for publication in the Astrophysical Journal