We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error correction criteria. Our tradeoff bounds relate the number of physical qubits, the number of encoded qubits, the code distance, the accuracy parameter that quantifies how well the erasure channel can be reversed, and the locality parameter that specifies the length scale at which the recovery operates. As a corollary, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the main assumptions of algebraic anyon theories, that there exist only finitely many anyon types.