We study the volume-limited and nearly mass-selected (stellar mass Mstars ≳ 6 × 109 M⊙) ATLAS3D sample of 260 early-type galaxies (ETGs, ellipticals Es and lenticulars S0s). We construct detailed axisymmetric dynamical models (Jeans Anisotropic MGE), which allow for orbital anisotropy, include a dark matter halo and reproduce in detail both the galaxy images and the high-quality integral-field stellar kinematics out to about 1Re, the projected half-light radius. We derive accurate total mass-to-light ratios (M/L)e and dark matter fractions fDM, within a sphere of radius r = Re centred on the galaxies.We alsomeasure the stellar (M/L)stars and derive a median dark matter fraction fDM = 13 per cent in our sample. We infer masses MJAM = L × (M/L)e ≈ 2 ×M1/2, where M1/2 is the total mass within a sphere enclosing half of the galaxy light. We find that the thin two-dimensional subset spanned by galaxies in the (MJAM, σe,Rmaje ) coordinates system, which we call the Mass Plane (MP) has an observed rms scatter of 19 per cent, which implies an intrinsic one of 11 per cent. Here, Rmaje is the major axis of an isophote enclosing half of the observed galaxy light, while σe is measuredwithin that isophote. The MP satisfies the scalar virial relation MJAM ∝ σ2e Rmaje within our tight errors. This show that the larger scatter in the Fundamental Plane (FP) (L, σe, Re) is due to stellar population effects [including trends in the stellar initial mass function (IMF)]. It confirms that the FP deviation from the virial exponents is due to a genuine (M/L)e variation. However, the details of how both Re and σe are determined are critical in defining the precise deviation from the virial exponents. The main uncertainty in masses or M/L estimates using the scalar virial relation is in the measurement of Re. This problem is already relevant for nearby galaxies and may cause significant biases in virial mass and size determinations at high redshift. Dynamical models can eliminate these problems.We revisit the (M/L)e-σe relation, which describes most of the deviations between the MP and the FP. The best-fitting relation is (M/L)e ∝ σ0.72e (r band). It provides an upper limit to any systematic increase of the IMF mass normalization with σe. The correlation is more shallow and has smaller scatter for slow rotating systems or for galaxies in Virgo. For the latter, when using the best distance estimates, we observe a scatter in (M/L)e of 11 per cent, and infer an intrinsic one of 8 per cent. We perform an accurate empirical study of the link between se and the galaxies circular velocity Vcirc within 1Re (where stars dominate) and find the relation max (Vcirc) ≈ 1.76 × σe, which has an observed scatter of 7 per cent. The accurate parameters described in this paper are used in the companion Paper XX (Cappellari et al.) of this series to explore the variation of global galaxy properties, including the IMF, on the projections of the MP.