We present high-resolution soft-X-ray spectra of the prototypical Seyfert 2 galaxy, NGC 1068, taken with XMM-Newton RGS and Chandra LETGS. Its rich emission-line spectrum is dominated by recombination in a warm plasma (bright, narrow radiative recombination continua provide the ``smoking gun), which is photoionized by the inferred nuclear power-law continuum. Radiative decay following photoexcitation of resonant transitions is also significant. A self-consistent model of an irradiated cone of gas is capable of reproducing the hydrogenic/heliumlike ionic line series in detail. The radial ionic column densities we infer are consistent with absorption measurements (the warm absorber) in Seyfert 1 galaxies. This strongly suggests that the emission spectrum we observe from NGC 1068 emanates from its warm absorber. The observed extent of the ionization-cone/warm absorber in NGC 1068 of about 300 pc implies that a large fraction of the gas associated with generic warm absorbers may typically exist on the hundreds-of-parsec scale rather than much closer to the nucleus (e.g., less than a parsec). Spatially-resolved spectroscopy using the LETGS of two distinct emission regions yields two noticeably different spectra. We show that these differences are solely due to differing radial column densities. A fairly flat distribution in ionization parameter is necessary to explain the inferred radial ionic column densities of all spectra. This must primarily be due to a broad density distribution at each radius, spanning roughly 0.1-100 cm$^{-3}$. (Abridged)