We report the confirmation of the binary nature of the nearby, very low-mass system NLTT 33370 with adaptive optics imaging and present resolved near-infrared photometry and integrated light optical and near-infrared spectroscopy to characterize the system. VLT-NaCo and LBTI-LMIRCam images show significant orbital motion between 2013 February and 2013 April. Optical spectra reveal weak, gravity sensitive alkali lines and strong lithium 6708 Angstrom absorption that indicate the system is younger than field age. VLT-SINFONI near-IR spectra also show weak, gravity sensitive features and spectral morphology that is consistent with other young, very low-mass dwarfs. We combine the constraints from all age diagnostics to estimate a system age of ~30-200 Myr. The 1.2-4.7 micron spectral energy distribution of the components point toward T_eff=3200 +/- 500 K and T_eff=3100 +/- 500 K for NLTT 33370 A and B, respectively. The observed spectra, derived temperatures, and estimated age combine to constrain the component spectral types to the range M6-M8. Evolutionary models predict masses of 113 +/- 8 M_Jup and 106 +/- 7 M_Jup from the estimated luminosities of the components. KPNO-Phoenix spectra allow us to estimate the systemic radial velocity of the binary. The Galactic kinematics of NLTT 33370AB are broadly consistent with other young stars in the Solar neighborhood. However, definitive membership in a young, kinematic group cannot be assigned at this time and further follow-up observations are necessary to fully constrain the system's kinematics. The proximity, age, and late-spectral type of this binary make it very novel and an ideal target for rapid, complete orbit determination. The system is one of only a few model calibration benchmarks at young ages and very low-masses. ; Comment: 25 pages, 3 tables, 13 figures, accepted for publication in The Astrophysical Journal