Given the high throughput requirement for the next generation wireless communication systems, merging millimeter wave technologies and multi-user MIMO seems a very promising strategy to achieve the required 20 Gbps. Although a full digital architecture provides the best performance and flexibility, its implementation at millimeter-wave frequencies today seems unrealistic due to the prohibitive costs and high power consumption. Hybrid analog-digital architectures, efficiently sharing beamforming operations between analog and digital domains, appear as a feasible way to implement multi-user MIMO systems at millimeter wave frequencies. While hybrid architectures have been studied intensively, an effective and flexible demonstration proving the feasibility is still missing. In this paper, we introduce the design of a millimeter wave multi-user MIMO system with hybrid analog and digital processing using 60 GHz radios equipped with phased arrays. A base station, employing two 16-antennas transmitters, serves simultaneously two user equipment devices, each with 4 antenna elements, effectively realizing spatial multiplexing. We propose a complete system design comprising the description of millimeter radio transceivers, the multi-user hybrid MIMO algorithms including a strategy for channel estimation and frequency selective precoding along with the transmission protocol. We show that the spatial multiplexing is achieved in several scenarios, most importantly in a strong interference-limited scenario.