Giant unilamellar vesicles (GUVs) provide a direct connection between the nano- and the microregime. On the one hand, these vesicles represent biomimetic compartments with linear dimensions of many micrometers. On the other hand, the vesicle walls are provided by single molecular bilayers that have a thickness of a few nanometers and respond sensitively to molecular interactions with small solutes, biopolymers, and nanoparticles. These nanoscopic responses are amplified by the GUVs and can then be studied on much larger scales. Therefore, GUVs are increasingly used as a versatile research tool for basic membrane science, bioengineering, and synthetic biology. Conventional GUVs have one major drawback, however: they have only a limited capability to cope with external perturbations such as osmotic inflation, adhesion, or micropipette aspiration that tend to rupture the membranes. In contrast, cell membranes tolerate the same kinds of mechanical perturbations without rupture because the latter membranes are coupled to reservoirs of membrane area. Here, we introduce GUVs with membrane nanotubes as model systems that include such area reservoirs. To demonstrate the increased robustness of these tubulated vesicles, we use micropipette aspiration and changes in the osmotic conditions applied to phospholipid membranes doped with the glycolipid GM1. A quantitative comparison between theory and experiment reveals that the response of the GUVs is governed by the membranes' spontaneous tension, a curvature-elastic material parameter that describes the bilayer asymmetry on the nanoscale. Because of their increased robustness, GUVs with nanotubes represent improved research tools for membrane science, in general, with potential applications as storage and delivery systems and as cell-like microcompartments in bioengineering, pharmacology, and synthetic biology.