We present the TESS discovery of the LHS 1678 (TOI-696) exoplanet system, comprised of two sub-Earth-sized transiting planets and a likely astrometric brown dwarf orbiting a bright M2 dwarf at 19.89 pc. The two TESS-detected planets are of radius 0.7 Earth radii and 0.9 Earth radii in 0.86-day and 3.69-day orbits, respectively. Both planets are confirmed and characterized via ground-based follow-up observations, including HARPS RV monitoring that yields 3-sigma mass upper limits. The astrometric companion detected by the SMARTS/CTIO 0.9m has an orbital period on the order of decades and is undetected by other means. Additional ground-based observations constrain the companion to early brown dwarf mass or smaller. Both planets are of unique interest; the inner planet is a rare ultrashort-period planet and a favorable target for emission spectroscopy observations with JWST. The outer planet is a Venus-zone planet receiving 13.5 times the flux that Earth receives from the Sun. Both planets are promising targets for JWST transmission spectroscopy measurements to characterize their atmospheres and for mass measurements via extreme precision radial velocity. Simulations indicate that the planets are dynamically stable over long time scales and their period separation allows for additional, yet-undetected planets. The host star itself is part of an older population and the only TESS host star to date located in a gap in the lower main sequence. This gap is tied to the transition between partial and full convection in M dwarfs, and the effect of the associated stellar astrophysics on exoplanet evolution is currently unknown. The culmination of these system properties makes LHS 1678 a unique, compelling playground for comparative exoplanet science and understanding the formation and evolution of small, short-period exoplanets orbiting low-mass stars.