We present new Chandra X-Ray Observatory observations of the L1551 molecular cloud in Taurus. We find a compact, but slightly resolved, X-ray source, displaced westward from the IRS 5 binary protostar by about 50-100 AU, which coincides with the base of the HH 154 protostellar jet. The column density of material lying in front of the X-ray source is more than an order of magnitude lower than the column density toward L1551 IRS 5. Thus, it is highly improbable that the X-rays come directly from the embedded protobinary. It is possible, however, that X-rays produced by one or both members of the IRS 5 binary escape through the outflow cavity and are scattered into the line of sight by a dense infalling envelope or material in the outflow. Constraints on the physical properties of the scattering medium are discussed. It is also possible that the X-rays are produced in situ by fast shocks at the base of the HH 154 jet. We consider several possible geometries for such shocks. The radiating plasma may be located behind standing shocks, where a wide-angle wind from one member of the IRS 5 binary is collimated into a jet, or behind shocks formed on the axis of the outflow, where winds from each member of the binary collide or where these winds impact a tilted circumbinary disk. To produce the observed X-ray luminosity and plasma temperature, shock velocities larger than 350 km s^-1 are required, with a preshock hydrogen density of (1–10) × 10³ cm^-3. The implied postshock cooling length is around 800 AU, close to the observed length of the bright near-infrared [Fe II] emission in the inner portion of the HH 154 jet.