We report X-ray imaging of the powerful FR-II radio galaxy 3C353 using the Chandra X-ray Observatory. 3C353s 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. In a 90 ks Chandra observation, we have detected X-ray emission from most radio structures in 3C353, 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 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.