We present a detailed spectral and spatial analysis of the X-ray properties of the compact and unusual radio galaxy 3C 346, combining information from ROSAT and ASCA. The dominant component of X-ray emission (∼1044 ergs s-1 in each of the 0.5-3 keV and 2-10 keV bands) is unresolved and not heavily absorbed (intrinsic NH≲2x1021 cm–2), with evidence for variability of 32%±13% over 18 months. We relate the X-ray emission to radio structures on both milliarcsecond scales and the arcsecond scales which Chandra can resolve. The absence of X-ray absorption, and the radio/optical/X-ray colors, when combined with previous radio evidence that the source is a foreshortened FR II, suggest that the radio jets are seen at an angle to the line of sight of about 30°, intermediate between the radio galaxy and quasar classes. Roughly a third of the soft X-ray emission is from a cluster atmosphere, for which we measure a temperature of 1.9+1.3–0.7 keV, making this the second low-redshift (z<0.2) powerful radio galaxy, after Cyg A, with a measured cluster temperature. At a jet angle of ∼30°, all the radio structures lie within the core radius of the cluster, for which the cooling time is sufficiently long that there is no reason to expect the presence of a cooling flow. The radio lobes of 3C 346 are roughly in pressure balance with the external medium under the assumptions that the energy densities in the magnetic field and radiating particles balance and that a source of excess pressure in the radio lobes, commonly invoked in other radio galaxies, is absent here.