One of the grand challenges in cellular biophysics is understanding the precision with which cells assemble and maintain subcellular structures. Organelle sizes, for example, must be flexible enough to allow cells to grow or shrink them as environments demand yet be maintained within homeostatic limits. Despite identification of molecular factors that regulate organelle sizes we lack insight into the quantitative principles underlying organelle size control. Here we show experimentally that cells can robustly control average fluctuations in organelle size. By demonstrating that organelle sizes obey a universal scaling relationship we predict theoretically, our framework suggests that organelles grow in random bursts from a limiting pool of building blocks. Burstlike growth provides a general biophysical mechanism by which cells can maintain on average reliable yet plastic organelle sizes.