Owing to the ubiquity of cellular communication signals, positioning with the fifth generation (5G) signal has emerged as a promising solution in global navigation satellite system-denied areas. Unfortunately, although the widely employed antenna arrays in 5G remote radio units (RRUs) facilitate the measurement of the direction of arrival (DOA), DOA-based positioning performance is severely degraded by array errors. This paper proposes an in-situ calibration framework with a user terminal transmitting 5G reference signals at several known positions in the actual operating environment and the accessible RRUs estimating their array errors from these reference signals. Further, since sub-6GHz small-cell RRUs deployed for indoor coverage generally have small-aperture antenna arrays, while 5G signals have plentiful bandwidth resources, this work segregates the multipath components via super-resolution delay estimation based on the maximum likelihood criteria. This differs significantly from existing in-situ calibration works which resolve multipaths in the spatial domain. The superiority of the proposed method is first verified by numerical simulations. We then demonstrate via field test with commercial 5G equipment that, a reduction of 46.7% for 1-$σ$ DOA estimation error can be achieved by in-situ calibration using the proposed method.