The molecular dynamics method is employed to study hydrates of methane (sI), and krypton hydrate (sII), as well as an ice nanocluster in a supercooled water shell. The main attention is focused on the local structure and the mechanical state of two-phase nanosized systems, which is described using the local pressure tensor. Analysis of the temperature dependence of the local pressure allows one to compare two possible mechanisms responsible for the anomalous stability of gas hydrates at ambient pressure. According to the first mechanism, the water shell plays the role of a barrier that prevents the gas from escaping from the hydrate core. The second mechanism implies that the water shell generates additional pressure, which transfers the hydrate to a thermodynamically stable state. Results of molecular dynamics simulation indicate that both mechanisms are simultaneously involved in the stabilization of the hydrate nanocluster.