We report the first measurements of the intrinsic strain fluctuations of living cells using a recently developed tracer correlation technique along with a theoretical framework for interpreting such data in heterogeneous media with nonthermal driving. The fluctuations' spatial and temporal correlations indicate that the cytoskeleton can be treated as a course-grained continuum with power-law rheology, driven by a spatially random stress tensor field. Combined with recent cell rheology results, our data imply that intracellular stress fluctuations have a nearly 1/omega2 power spectrum, as expected for a continuum with a slowly evolving internal prestress.