Modern mobile terminals produce massive small data packets. For these short-length packets, it is inefficient to follow the current multiple access schemes to allocate transmission resources due to heavy signaling overhead. We propose a non-orthogonal many-access scheme that is well suited for the future communication systems equipped with many receive antennas. The system is modeled as having a block-sparsity pattern with unknown sparsity level (i.e., unknown number of transmitted messages). Block precoding is employed at each single-antenna transmitter to enable the simultaneous transmissions of many users. The number of simultaneously served active users is allowed to be even more than the number of receive antennas. Sparsity-aware recovery is designed at the receiver for joint user detection and symbol demodulation. To reduce the effects of channel fading on signal recovery, normalized block orthogonal matching pursuit (BOMP) algorithm is introduced, and based on its approximate performance analysis, we develop interference cancellation-based BOMP (ICBOMP) algorithm. The ICBOMP performs error correction and detection in each iteration of the normalized BOMP. Simulation results demonstrate the effectiveness of the proposed scheme in small packet services, as well as the advantages of ICBOMP in improving signal recovery accuracy and reducing computational cost.