Among the most challenging traffic-analysis attacks to confound are those leveraging the sizes of objects downloaded over the network. In this paper we systematically analyze this problem under realistic constraints regarding the padding overhead that the object store is willing to incur. We give algorithms to compute privacy-optimal padding schemes -- specifically that minimize the network observer's information gain from a downloaded object's padded size -- in several scenarios of interest: per-object padding, in which the object store responds to each request for an object with the same padded copy; per-request padding, in which the object store pads an object anew each time it serves that object; and a scenario unlike the previous ones in that the object store is unable to leverage a known distribution over the object queries. We provide constructions for privacy-optimal padding in each case, compare them to recent contenders in the research literature, and evaluate their performance on practical datasets.