This paper investigates a hardware-efficient massive multiple-input multiple-output integrated sensing and communication (MIMO-ISAC) system with 1-bit analog-to-digital converters (ADCs)/digital-to-analog converters (DACs). The proposed system, referred to as 1BitISAC, employs 1-bit DACs at the ISAC transmitter and 1-bit ADCs at the sensing receiver, achieving significant reductions in power consumption and hardware costs. For such kind of systems, two 1BitISAC joint transceiver designs, i.e., i) quality of service constrained 1BitISAC design and ii) quality of detection constrained design, are considered and the corresponding problems are formulated. In order to address these problems, we thoroughly analyze the radar detection performance after 1-bit ADCs quantization and the communication bit error rate. This analysis yields new design insights and leads to unique radar and communication metrics, which enables us to simplify the original problems and employ majorization-minimization and integer linear programming methods to solve the problems. Numerical results are provided to validate the performance analysis of the proposed 1BitISAC and to compare with other ISAC configurations. The superiority of the proposed 1BitISAC system in terms of balancing ISAC performance and energy efficiency is also demonstrated.