  In a companion paper we have reported a $>5\sigma$ detection of degree scale
$B $-mode polarization at 150 GHz by the BICEP2 experiment. Here we provide a
detailed study of potential instrumental systematic contamination to that
measurement. We focus extensively on spurious polarization that can potentially
arise from beam imperfections. We present a heuristic classification of beam
imperfections according to their symmetries and uniformities, and discuss how
resulting contamination adds or cancels in maps that combine observations made
at multiple orientations of the telescope about its boresight axis. We
introduce a technique, which we call "deprojection", for filtering the leading
order beam-induced contamination from time ordered data, and show that it
removes power from BICEP2's $BB$ spectrum consistent with predictions using
high signal-to-noise beam shape measurements. We detail the simulation pipeline
that we use to directly simulate instrumental systematics and the calibration
data used as input to that pipeline. Finally, we present the constraints on
$BB$ contamination from individual sources of potential systematics. We find
that systematics contribute $BB$ power that is a factor $\sim10\times$ below
BICEP2's 3-year statistical uncertainty, and negligible compared to the
observed $BB$ signal. The contribution to the best-fit tensor/scalar ratio is
at a level equivalent to $r=(3-6)\times10^{-3}$.
