Cells can precisely program the shape and lateral organization of their membranes using protein machinery. Aiming to replicate a comparable degree of control, here we introduce DNA-origami line-actants (DOLAs) as synthetic analogues of membrane-sculpting proteins. DOLAs are designed to selectively accumulate at the line-interface between coexisting domains in phase-separated lipid membranes, modulating the tendency of the domains to coalesce. With experiments and coarse-grained simulations, we demonstrate that DOLAs can reversibly stabilize two-dimensional analogues of Pickering emulsions on synthetic giant liposomes, enabling dynamic programming of membrane lateral organization. The control afforded over membrane structure by DOLAs extends to three-dimensional morphology, as exemplified by a proof-of-concept synthetic pathway leading to vesicle fission. With DOLAs we lay the foundations for mimicking, in synthetic systems, some of the critical membrane-hosted functionalities of biological cells, including signaling, trafficking, sensing, and division.