This paper studies a majority binary neural network characterized by simple majority neurons from three binary inputs to one binary output. The dynamics is described by an autonomous difference equation and the network exhibits various binary periodic orbits. In order to analyze the network, we present simple feature quantities that evaluate period length and direct stability of periodic orbits (fixed points). Performing precise numerical analysis for low-dimensional networks, we have clarified that the networks exhibit a variety of BPOs having a long period and/or strong direct stability. Also, we have discovered multiple fixed points with the strongest direct stability. As a fundamental step to engineering applications, an FPGA-based hardware prototype is presented and typical periodic orbits are experimentally confirmed.