Although the current H1N1 influenza strain is now considered to be relatively mild, it still has the potential to place a serious burden on health services. The closure of schools has been postulated as a means of reducing transmission between children and hence reducing the number of cases at the peak of an epidemic; however if instigated nationally such a policy has severe economic costs. Reactive short-duration closure of schools in regions where health services are close to capacity offers a potential compromise, but it is unclear over what spatial scale and timeframe closures would need to be made to have a substantial impact. Here, using detailed geographic information for England, we assess how localized school closures could alleviate the burden on hospital intensive care units (ICUs) that are reaching capacity. We show that, for a range of epidemiologically plausible assumptions, considerable local coordination of school closures is needed to achieve a substantial reduction in the number of hospitals that are over capacity at the epidemic peak. The heterogeneity in demand per hospital means that even widespread school closures are unlikely to impact on whether demand will exceed capacity for many hospital ICUs. These results re-enforce the UK policy of not utilising school closures as a control mechanism, but have far wider international public-health implications. The spatial heterogeneities in both population density and hospital capacity that give rise to our results are present in many Northern Hemisphere countries where a second wave of influenza is predicted this autumn and winter. This leads us to believe that even widespread reactive school closures are unlikely to eliminate problems of demand exceeding local capacity in many regions.