Methane hydrate is widely distributed in the pores of marine sediments or permafrost soils, contributing to their mechanical properties. Yet the tensile properties of the hydrate at pore scales remain almost completely unknown, notably the influence of grain size on its own cohesion. Here we grow thin films of the hydrate in glass capillaries. Using a novel, contactless thermal method to apply stress, and video microscopy to observe the strain, we estimate the tensile elastic modulus and strength. Ductile and brittle characteristics are both found, dependent on sample thickness and texture, which are controlled by supercooling with respect to the dissociation temperature and by ageing. Relating the data to the literature suggests the cohesive strength of methane hydrate was so far significantly overestimated.