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We report results of a sensitive Chandra ACIS-S observation of the classical T Tauri star RY Tau. Previous studies have shown that it drives a spectacular bipolar jet whose blueshifted component is traced optically along P.A. approximately 295 degrees at separations of 1.5 - 31 arcseconds from the star. Complex X-ray emission is revealed, including a very soft non-variable spectral component (some of which may originate in shocks), a superhot flaring component (T >= 100 MK), and faint extended structure near the star. The structure is visible in deconvolved images and extends northwestward out to a separation of 1.7 arcseconds, overlapping the inner part of the optical jet. Image analysis suggests that most of the extension is real, but some contamination by PSF-induced structure within the central arcsecond may be present. The predicted temperature for a shock-heated jet based on jet speed and shock speed estimates from optical measurements is too low to explain the extended X-ray structure. Either higher speed material within the jet has escaped optical detection or other mechanisms besides shock-heating are involved. Alternative mechanisms that could produce higher temperature plasma at small offsets to the northwest of RY Tau include magnetic heating in the jet, hot plasmoids ejected at high speeds, or X-ray emission from a putative close companion whose presence has been inferred from Hipparcos variations.