  We present near-infrared spectroscopy of a sample of 22 Extreme Emission Line
Galaxies at redshifts 1.3 < z < 2.3, confirming that these are low-mass (M* =
10^8 - 10^9 M_sun) galaxies undergoing intense starburst episodes (M*/SFR ~
10-100 Myr). The sample is selected by [O III] or H{\alpha} emission line flux
and equivalent width using near-infrared grism spectroscopy from the 3D-HST
survey. High-resolution NIR spectroscopy is obtained with LBT/LUCI and
VLT/X-SHOOTER. The [O III]/H{\beta} line ratio is high (> 5) and [N
II]/H{\alpha} is always significantly below unity, which suggests a low
gas-phase metallicity. We are able to determine gas-phase metallicities for 7
of our objects using various strong-line methods, with values in the range
0.05-0.30 Z_sun and with a median of 0.15 Z_sun; for 3 of these objects we
detect [O III]{\lambda}4363 which allows for a direct constraint on the
metallicity. The velocity dispersion, as measured from the nebular emission
lines, is typically ~50 km/s. Combined with the observed star-forming activity,
the Jeans and Toomre stability criteria imply that the gas fraction must be
large (> 2/3), consistent with the difference between our dynamical and stellar
mass estimates. The implied gas depletion time scale (several hundred Myr) is
substantially longer than the inferred mass-weighted ages (~50 Myr), which
further supports the emerging picture that most stars in low-mass galaxies form
in short, intense bursts of star formation.
