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We present results on the nature of extreme ejective feedback episodes and the physical conditions of a population of massive (M_*∼ 10¹¹M_⊙), compact starburst galaxies atz= 0.4–0.7. We use data from Keck/NIRSPEC, SDSS, Gemini/GMOS, MMT, and Magellan/MagE to measure rest-frame optical and near-IR spectra of 14 starburst galaxies with extremely high star formation rate surface densities (mean Σ_SFR∼ 2000M_⊙yr⁻¹kpc⁻²) and powerful galactic outflows (maximum speedsv₉₈∼ 1000–3000 km s⁻¹). Our unique data set includes an ensemble of both emission ([Oii]λλ3726,3729, Hβ, [Oiii]λλ4959,5007, Hα, [Nii]λλ6549,6585, and [Sii]λλ6716,6731) and absorption (Mgiiλλ2796,2803, and Feiiλ2586) lines that allow us to investigate the kinematics of the cool gas phase (T∼ 10⁴K) in the outflows. Employing a suite of line ratio diagnostic diagrams, we find that the central starbursts are characterized by high electron densities (mediann_e∼ 530 cm⁻³), and high metallicity (solar or supersolar). We show that the outflows are most likely driven by stellar feedback emerging from the extreme central starburst, rather than by an AGN. We also present multiple intriguing observational signatures suggesting that these galaxies may have substantial Lyman continuum (LyC) photon leakage, including weak [Sii]nebular emission lines. Our results imply that these galaxies may be captured in a short-lived phase of extreme star formation and feedback where much of their gas is violently blown out by powerful outflows that open up channels for LyC photons to escape.