We have obtained and analyzed Chandra ACIS-S observations of the strongly accreting classical T Tauri star DG Tau. Our principal goals are to map the immediate environment of the star to characterize possible extended X-rays formed in the jet, and to re-visit the anomalous, doubly absorbed X-ray spectrum of DG Tau itself. We combine our new ACIS-S data with a data set previously obtained. The data are superimposed to obtain flux and hardness images. Separate X-ray spectra are extracted for DG Tau and areas outside its point spread function. We detect a prominent X-ray jet at a position angle of PA ~225 deg (tentatively suggested by Guedel et al. 2005), coincident with the optical jet axis. We also identify a counter jet at PA = 45 deg. The X-ray jets are detected out to a distance of ~5' from the star, their sources being extended at the ACIS-S resolution. The jet spectra are soft, with a best-fit electron temperature of 3.4 MK. We find evidence for excess absorption of the counter jet. The spectrum of the DG Tau point source shows two components with largely different temperatures and absorption column densities. The similar temperatures and small absorbing gas columns of the jet sources and the soft component of the 'stellar' source suggest that these sources are related, produced either by shocks or by magnetic heating in the jets. Cooling estimates suggest that the pressure in the hot gas contributes to jet expansion. The hard 'stellar' component, on the other hand, is associated with a stellar corona or magnetosphere. The excessive photoelectric absorption of this component suggests the presence of dust-depleted accretion streams above coronal magnetic fields.