[Abridged] We report observations of the recently discovered, nearby late-M dwarf WISE J072003.20-084651.2. Astrometric measurements obtained with TRAPPIST improve the distance measurement to 6.0$pm$1.0 pc and confirm the low tangential velocity (3.5$pm$0.6 km/s) reported by Scholz. Low-resolution optical spectroscopy indicates a spectral type of M9.5 and prominent H$alpha$ emission (<LH$alpha$/Lbol> = -4.68$pm$0.06), but no evidence of subsolar metallicity or Li I absorption. Near-infrared spectroscopy reveals subtle peculiarities indicating the presence of a T5 binary companion, and high-resolution laser guide star adaptive optics imaging reveals a faint ($Delta$H = 4.1) candidate source 014 (0.8 AU) from the primary. We measure a stable radial velocity of +83.8$pm$0.3 km/s, indicative of old disk kinematics and consistent with the angular separation of the possible companion. We measure a projected rotational velocity of v sin i = 8.0$pm$0.5 km/s, and find evidence of low-level variability (~1.5%) in a 13-day TRAPPIST lightcurve, but cannot robustly constrain the rotational period. We also observe episodic changes in brightness (1-2%) and occasional flare bursts (4-8%) with a 0.8% duty cycle, and order-of-magnitude variations in H$alpha$ line strength. Combined, these observations reveal WISE J0720-0846 to be an old, very low-mass binary whose components straddle the hydrogen burning minimum mass, and whose primary is a relatively rapid rotator and magnetically active. It is one of only two known binaries among late M dwarfs within 10 pc of the Sun, both harboring a mid T-type brown dwarf companion. While this specific configuration is rare (1.4% probability), roughly 25% of binary companions to late-type M dwarfs in the local population are likely low-temperature T or Y brown dwarfs.
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