We report X-ray imaging of the powerful FR II radio galaxy 3C 353 using the Chandra X-ray Observatory. 3C 353's two 4 ′′ -wide and 2 ′ -long jets allow us to study in detail the internal structure of the large-scale relativistic outflows at both radio and X-ray photon energies with the sub-arcsecond spatial resolution provided by the VLA and Chandra instruments. In a 90 ks Chandra observation, we have detected X-ray emission from most radio structures in 3C 353, including the nucleus, the jet and the counterjet, the terminal jet regions (hotspots), and one radio lobe. We show that the detection of the X-ray emission associated with the radio knots and counterknots, which is most likely non-thermal in origin, puts several crucial constraints on the X-ray emission mechanisms in powerful large-scale jets of quasars and FR II sources. In particular, we show that this detection is inconsistent with the inverse-Compton model proposed in the literature, and instead implies a synchrotron origin of the X-ray jet photons. We also find that the width of the X-ray counterjet is possibly narrower than that measured in radio bands, that the radio-to-X-ray flux ratio decreases systematically downstream along the jets, and that there are substantial (kpc-scale) offsets between the positions of the X-ray and radio intensity maxima within each knot, whose magnitudes increase away from the nucleus. We discuss all these findings in the wider context of the physics of extragalactic jets, proposing some particular though not definitive solutions or interpretations for each problem. In general, we find that the synchrotron X-ray emission of extragalactic large-scale jets is not only shaped by the global hydrodynamical configuration of the outflows, but is also likely to be very sensitive to the microscopic parameters of the jet plasma. A complete, self-consistent model for the X-ray emission of extragalactic jets still remains elusive.