We show that every symmetric normed space admits an efficient nearest neighbor search data structure with doubly-logarithmic approximation. Specifically, for every (n), (d = n^{o(1)}), and every (d)-dimensional symmetric norm (|\cdot|), there exists a data structure for (\mathrm{poly}(log log n))-approximate nearest neighbor search over (|\cdot|) for (n)-point datasets achieving (n^{o(1)}) query time and (n^{1+o(1)}) space. The main technical ingredient of the algorithm is a low-distortion embedding of a symmetric norm into a low-dimensional iterated product of top-(k) norms. We also show that our techniques cannot be extended to general norms.