  We report on a search for electron antineutrinos ($\bar{\nu}_e$) from
astrophysical sources in the neutrino energy range 8.3 to 30.8 MeV with the
KamLAND detector. In an exposure of 6.72 kton-year of the liquid scintillator,
we observe 18 candidate events via the inverse beta decay reaction. Although
there is a large background uncertainty from neutral current atmospheric
neutrino interactions, we find no significant excess over background model
predictions. Assuming several supernova relic neutrino spectra, we give upper
flux limits of 60--110 cm$^{-2}$ s$^{-1}$ (90% CL) in the analysis range and
present a model-independent flux. We also set limits on the annihilation rates
for light dark matter pairs to neutrino pairs. These data improves on the upper
probability limit of $^{8}$B solar neutrinos converting into $\bar{\nu}_e$'s,
$P_{\nu_e \rightarrow \bar{\nu}_e} < 3.5\times10^{-5}$ (90% CL) assuming an
undistorted $\bar{\nu}_e$ shape. This corresponds to a solar $\bar{\nu}_e$ flux
of 60 cm$^{-2}$ s$^{-1}$ (90% CL) in the analysis energy range.
