Nanobubbles filled with air or a variety of pure gases are observed to persist in bulk water for weeks and months. Nanoemulsions consisting of oil droplets in water are also remarkably stable against coagulation, with lifetimes up to weeks even if not coated with surfactants. The inverse system of nanodroplets of water in oil is also accessible for study and application. Voids on the nanoscale are formed in simulations of water under strong tension and are stable during the time of the simulation. The stability of these nano-entities is ultimately determined by the molecular-level structure of their interfaces. However, a thermodynamic theory might also be capable of providing some insight. We therefore consider spherical gas nanobubbles, immiscible liquid nanodroplets, and nanocavities formed in water under negative pressure on the same footing, and give a unified thermodynamic analysis of these systems. In all cases, mechanical equilibrium (local free energy maximum or minimum) is expressed by the Laplace equation, and thermodynamic stability (local free energy minimum) follows from the radius dependence of surface tension. All of them would be unstable if their surface tensions were constant. Data from the literature allow construction of numerical examples for cavities and gas nanobubbles. Spectroscopic data are cited in support of an interfacial water structure in gas nanobubbles and water droplets in oil that differ from their flat surface counterparts. The observed longevity of nanobubbles in particular has been thought to violate fundamental principles of diffusion and solubility. A close look at the Laplace equation and its derivation shows why this widespread belief is incorrect.