We present Magellan/IMACS spectroscopy of the recently-discovered Milky Way satellite Eridanus II (Eri II). We identify 28 member stars in Eri II, from which we measure a systemic radial velocity of $v_{rm hel} = 75.6 pm 1.3~mbox{(stat.)} pm 2.0~mbox{(sys.)}~mathrm{km,s^{-1}}$ and a velocity dispersion of $6.9^{+1.2}_{-0.9}~mathrm{km,s^{-1}}$. Assuming that Eri~II is a dispersion-supported system in dynamical equilibrium, we derive a mass within the half-light radius of Eri II is $1.2^{+0.4}_{-0.3} times 10^{7}~mathrm{M_odot}$, indicating a mass-to-light ratio of $420^{+210}_{-140}~mathrm{M_odot}/mathrm{L_odot}$ and confirming that it is a dark matter-dominated dwarf galaxy. From the equivalent width measurements of the CaT lines of 16 red giant member stars, we derive a mean metallicity of ${rm [Fe/H]} = -2.38 pm 0.13$ and a metallicity dispersion of $sigma_{rm [Fe/H]} = 0.47 ^{+0.12}_{-0.09}$. The velocity of Eri II in the Galactic Standard of Rest frame is $v_{rm GSR} = -66.6~mathrm{km,s^{-1}}$, indicating that either Eri II is falling into the Milky Way potential for the first time or it has passed the apocenter of its orbit on a subsequent passage. At a Galactocentric distance of $sim$370 kpc, Eri II is one of the Milky Ways most distant satellites known. Additionally, we show that the bright blue stars previously suggested to be a young stellar population are not associated with Eri II. The lack of gas and recent star formation in Eri II is surprising given its mass and distance from the Milky Way, and may place constraints on models of quenching in dwarf galaxies and on the distribution of hot gas in the Milky Way halo. Furthermore, the large velocity dispersion of Eri II can be combined with the existence of a central star cluster to constrain MACHO dark matter with mass $gtrsim10~mathrm{M_odot}$.
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