We consider a distributed storage system where the storage nodes have heterogeneous access probabilities. The problem is to allocate a given storage budget across the nodes so as to store a unit-size data object with maximum reliability. We propose efficient algorithms for optimizing over several classes of allocations. In the basic one-level symmetric allocation, the storage budget is spread evenly over an appropriately chosen subset of nodes. In the multi k-level symmetric allocation, the budget is divided into k parts, each spread evenly over a different subset of nodes, such that the amount allocated to each node in the higher levels is multiple times that of the last level. These allocations are simpler and are shown to outperform existing methods in numerical experiments. We also describe an application of the symmetric allocations to the design of streaming codes.