  [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$H = 4.1) candidate source 0"14 (0.8 AU) from the primary. We measure a
stable radial velocity of +83.8$\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.
