DNA within cells is subject to damage from various sources. Organisms have evolved a number of mechanisms to repair DNA damage. The activity of repair enzymes carries its own risk, however, because the repair of two nearby lesions may lead to the breakup of DNA and result in cell death. We propose a mathematical theory of the damage and repair process in the important scenario where lesions are caused in bursts. We use this model to show that there is an optimum level of repair enzymes within cells which optimises the cell's response to damage. This optimal level is explained as the best trade-off between fast repair and a low probability of causing double-stranded breaks. We derive our results analytically and test them using stochastic simulations, and compare our predictions with current biological knowledge.