The detection of methane on Mars has revived the possibility of past or extant life on this planet, despite the fact that an abiogenic origin is thought to be equally plausible. An intriguing aspect of the recent observations of methane on Mars is that methane concentrations appear to be locally enhanced and change with the seasons. However, methane has a photochemical lifetime of several centuries, and is therefore expected to have a spatially uniform distribution on the planet. Here we use a global climate model of Mars with coupled chemistry to examine the implications of the recently observed variations of Martian methane for our understanding of the chemistry of methane. We find that photochemistry as currently understood does not produce measurable variations in methane concentrations, even in the case of a current, local and episodic methane release. In contrast, we find that the condensation-sublimation cycle of Mars' carbon dioxide atmosphere can generate large-scale methane variations differing from those observed. In order to reproduce local methane enhancements similar to those recently reported, we show that an atmospheric lifetime of less than 200 days is necessary, even if a local source of methane is only active around the time of the observation itself. This implies an unidentified methane loss process that is 600 times faster than predicted by standard photochemistry. The existence of such a fast loss in the Martian atmosphere is difficult to reconcile with the observed distribution of other trace gas species. In the case of a destruction mechanism only active at the surface of Mars, destruction of methane must occur with an even shorter timescale of the order of approximately 1 hour to explain the observations. If recent observations of spatial and temporal variations of methane are confirmed, this would suggest an extraordinarily harsh environment for the survival of organics on the planet.