In a companion paper we have reported a $>5sigma$ 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 BICEP2s $BB$ 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 $sim10times$ below BICEP2s 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 $r=(3-6)times10^{-3}$.
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