We analyze the large-scale structure and fluctuations of jammed packings of size-disperse spheres, produced in a granular experiment as well as numerically. While the structure factor of the packings reveals no unusual behavior for small wave vectors, the compressibility displays an anomalous linear dependence at low wave vectors and vanishes when q→0. We show that such behavior occurs because jammed packings of size-disperse spheres have no bulk fluctuations of the volume fraction and are thus hyperuniform, a property not observed experimentally before. Our results apply to arbitrary particle size distributions. For continuous distributions, we derive a perturbative expression for the compressibility that is accurate for polydispersity up to about 30%.