We derive a formula that defines quantum fluctuations of energy in subsystems of a hot relativistic gas. For small subsystem sizes, we find a substantial increase of fluctuations compared to those known from the standard thermodynamic considerations. However, if the size of the subsystem is sufficiently large, we reproduce the result for energy fluctuations in the canonical ensemble. Our results are subsequently used in the context of relativistic heavy-ion collisions to introduce limitations of the concepts such as classical energy density or fluid element. In a straightforward way, our formula can be applied in other fields of physics, wherever one deals with hot and relativistic matter.