We define and study a continuous version of 2-frieze patterns, a combinatorial structure closely related with frieze patterns of Coxeter and Conway. We describe the relation of continuous 2-friezes with the moduli space of projective curves and relate the (pre)symplectic structure on the space of closed 2-friezes, considered as a cluster variety, with the Adler-Gelfand-Dikii bracket on the space of 3rd order differential operators.