Drawing from the LEGA-C dataset, we present the spectroscopic view of the stellar population across a large volume- and mass-selected sample of galaxies at large lookback time. We measure the 4000\AA\ break (D$_n$4000) and Balmer absorption line strengths (probed by H$\delta$) from 1019 high-quality spectra of $z=0.6 - 1.0$ galaxies with $M_\ast = 2 \times 10^{10} M_\odot - 3 \times 10^{11} M_\odot$. Our analysis serves as a first illustration of the power of high-resolution, high-S/N continuum spectroscopy at intermediate redshifts as a qualitatively new tool to constrain galaxy formation models. The observed D$_n$4000-EW(H$\delta$) distribution of our sample overlaps with the distribution traced by present-day galaxies, but $z\sim 0.8$ galaxies populate that locus in a fundamentally different manner. While old galaxies dominate the present-day population at all stellar masses $> 2\times10^{10} M_\odot$, we see a bimodal D$_n$4000-EW(H$\delta$) distribution at $z\sim0.8$, implying a bimodal light-weighted age distribution. The light-weighted age depends strongly on stellar mass, with the most massive galaxies $>1\times10^{11}M_\odot$ being almost all older than 2 Gyr. At the same time we estimate that galaxies in this high mass range are only $\sim3$ Gyr younger than their $z\sim0.1$ counterparts, at odd with pure passive evolution given a difference in lookback time of $>5$ Gyr; younger galaxies must grow to $>10^{11}M_\odot$ in the meantime, and/or small amounts of young stars must keep the light-weighted ages young. Star-forming galaxies at $z\sim0.8$ have stronger H$\delta$ absorption than present-day galaxies with the same D$_n$4000, implying larger short-term variations in star-formation activity.