Massive multiple-input multiple-output (MIMO) is seen as an essential feature for the fifth-generation (5G) communication systems. To ensure the array performance for beamforming or nulling, it is important to conduct array calibration beforehand. Since 5G massive MIMO base stations (BSs) are expected to be highly integrated, i.e. the antenna arrays are directly integrated with the transceiver front ends, there may be no antenna connectors reserved for calibration purpose. In this case, array calibration will be done fully over-the-air (OTA). In this work, we propose a fast and short-distance OTA calibration method for massive MIMO BSs based on digital beamforming structures. Specifically, the correlation-based sounding technique is utilized for simultaneous measurements, and phase deviation due to the spherical wavefront at short distance is corrected with the known propagation geometry. Moreover, a multipath cancellation scheme is adopted to improve the calibration accuracy. An experiment is conducted in an open laboratory environment to assess the validity and robustness of the proposed calibration method.