A promising approach to the next generation of low-power, functional, and energy-efficient electronics relies on novel materials with coupled magnetic and electric degrees of freedom. In particular, stripy antiferromagnets often exhibit broken crystal and magnetic symmetries, which may bring about the magnetoelectric (ME) effect and enable the manipulation of intriguing properties and functionalities by electrical means. The demand for expanding the boundaries of data storage and processing technologies has led to the development of spintronics toward two-dimensional (2D) platforms. This work reports the ME effect in the 2D stripy antiferromagnetic insulator CrOCl down to a single layer. By measuring the tunneling resistance of CrOCl on the parameter space of temperature, magnetic field, and applied voltage, we verified the ME coupling down to the 2D limit and probed its mechanism. Utilizing the multi-stable states and ME coupling at magnetic phase transitions, we realize multi-state data storage in the tunneling devices. Our work not only advances the fundamental understanding of spin-charge coupling, but also demonstrates the great potential of 2D antiferromagnetic materials to deliver devices and circuits beyond the traditional binary operations.