We present spatially resolved stellar kinematics for 797 $z=0.6-1$ galaxies selected from the LEGA-C survey and construct axisymmetric Jeans models to quantify their dynamical mass and degree of rotational support. The survey is $K_s$-band selected, irrespective of color or morphological type, and allows for a first assessment of the stellar dynamical structure of the general $L^*$ galaxy population at large lookback time. Using light profiles from Hubble Space Telescope imaging as a tracer, our approach corrects for observational effects (seeing convolution and slit geometry), and uses well-informed priors on inclination, anisotropy and a non-luminous mass component. Tabulated data include total mass estimates in a series of spherical apertures (1, 5, and 10 kpc; 1$\times$ and 2$\times$\re), as well as rotational velocities, velocity dispersions and anisotropy. We show that almost all star-forming galaxies and $\sim$50\% of quiescent galaxies are rotation-dominated, with deprojected $V/\sigma\sim1-2$. Revealing the complexity in galaxy evolution, we find that the most massive star-forming galaxies are among the most rotation-dominated, and the most massive quiescent galaxies among the least rotation-dominated galaxies. These measurements set a new benchmark for studying galaxy evolution, using stellar dynamical structure for galaxies at large lookback time. Together with the additional information on stellar population properties from the LEGA-C spectra, the dynamical mass and $V/\sigma$ measurements presented here create new avenues for studying galaxy evolution at large lookback time.