The large-scale crossbar array is a promising architecture for hardware-amenable energy efficient three-dimensional memory and neuromorphic computing systems. While accessing a memory cell with negligible sneak currents remains a fundamental issue in the crossbar array architecture, up-to-date memory cells for large-scale crossbar arrays suffer from process and device integration (one selector one resistor) or destructive read operation (complementary resistive switching). Here, we introduce a self-selective memory cell based on hexagonal boron nitride and graphene in a vertical heterostructure. Combining non-volatile and volatile memory operations in the two hexagonal boron nitride layers, we demonstrate a self-selectivity of 1010 with an on/off resistance ratio larger than 103. The graphene layer efficiently blocks the diffusion of volatile silver filaments to integrate the volatile and non-volatile kinetics in a novel way. Our self-selective memory minimizes sneak currents on large-scale memory operation, thereby achieving a practical readout margin for terabit-scale and energy-efficient memory integration.