Giant unilamellar vesicles (GUVs) are microcompartments serving to confine reactions, allow signaling pathways, or design synthetic cells. Polymer GUVs are composed of copolymer membranes mimicking cell membranes, and present advantages over lipid-based GUVs, such as higher mechanical stability and chemical versatility. Such microcompartments are essential for understanding reactions/signaling occurring in cells, which are difficult to study by in vivo approaches due to the cell's complexity. However, the lack of control over their production, stability, and membrane diffusion properties is still limiting their use for bio-related applications. Here, polymer GUVs produced by microfluidics and permeabilized with DNA-origami nanopores (DoNs) that present a high level of control over these essential properties are introduced. After systematic optimization of conditions, DoN-GUVs reveal a narrow size distribution, allow for high encapsulation efficiencies, and are stable for weeks, protecting encapsulated biomolecules. The kinetics of diffusion of molecules through the GUV's membrane is tuned by insertion of DoNs with a controlled 3D- structure. DNA polymerase I, encapsulated as model for bioreactions, successfully produced DNA duplex strands with spatiotemporal control. DoN-GUVs loaded with active molecules open new avenues in bioreactions, from the detection of biomolecules, over the tuning of molecular transport rates, to the investigation of cellular processes/signaling.