The motion in the cytosol of microorganisms such as bacteria and yeast has been observed to undergo a dramatic slowing down upon cell energy depletion. These observations have been interpreted as the motion being "glassy", but whether this notion is useful also for active, motor protein-driven transport in eukaryotic cells is less clear. Here we use fluorescence microscopy of beads in human (HeLa) cells to probe the motion of membrane-surrounded structures that are carried along the cytoskeleton by motor proteins. Evaluating several hallmarks of glassy dynamics, we show that at short length scales the motion is heterogeneous, is non-ergodic, is well-described by a model for the displacement distribution in glassy systems and exhibits a decoupling of the exchange and persistence times. Overall, these results suggest that the short length scale behaviour of objects that can be transported actively by motor proteins in human cells shares features with the motion in glassy systems.