L 98-59 (TIC 307210830, TOI-175) is a nearby M3 dwarf around which TESS revealed three terrestrial-sized transiting planets (0.80, 1.35, 1.57 Earth radii) in a compact configuration with orbital periods shorter than 7.5 days. Here we aim to measure the masses of the known transiting planets in this system using precise radial velocity (RV) measurements taken with the HARPS spectrograph. We consider both trained and untrained Gaussian process regression models of stellar activity to simultaneously model the RV data with the planetary signals. Our RV analysis is then supplemented with dynamical simulations to provide strong constraints on the planets orbital eccentricities by requiring long-term stability. We measure the planet masses of the two outermost planets to be $2.46pm 0.31$ and $2.26pm 0.50$ Earth masses which confirms their bulk terrestrial compositions. We are able to place an upper limit on the mass of the smallest, innermost planet of $<0.98$ Earth masses with 95% confidence. Our RV plus dynamical stability analysis places strong constraints on the orbital eccentricities and reveals that each planets orbit likely has $e<0.1$ to ensure a dynamically stable system. The L 98-59 compact system of three likely rocky planets offers a unique laboratory for studies of planet formation, dynamical stability, and comparative atmospheric planetology. Continued RV monitoring will help refine the characterization of the innermost planet and potentially reveal additional planets in the system at wider separations.
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