We report observations of the recently discovered, nearby late-M dwarf WISE J072003.20-084651.2. New astrometric measurements obtained with the TRAPPIST telescope improve the distance measurement to 6.0 ± 1.0 pc and confirm the low tangential velocity (3.5 ± 0.6 km s‑1) reported by Scholz. Low-resolution optical spectroscopy indicates a spectral type of M9.5 and prominent Hα emission (< {{log }10}{{L}Hα }/{{L}bol}> = ‑4.68 ± 0.06), but no evidence of subsolar metallicity or Li i absorption. Near-infrared spectroscopy reveals subtle peculiarities that can be explained by the presence of a T5 binary companion, and high-resolution laser guide star adaptive optics imaging reveals a faint (ΔH = 4.1) candidate source 0\buildrel{\prime\prime}\over{.} 14 (0.8 AU) from the primary. With high-resolution optical and near-infrared spectroscopy, we measure a stable radial velocity of +83.8 ± 0.3 km s‑1, 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 ± 0.5 km s‑1 and find evidence of low-level variabilty (∼1.5%) in a 13 day TRAPPIST light curve, 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 Hα 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 of which harbor a mid T-type brown dwarf companion. We show that while this specific configuration is rare (≲1.6% probability), roughly 25% of binary companions to late-type M dwarfs in the local population are likely low-temperature T or Y brown dwarfs. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.