Mechanics of living cell interior are governed by cytoskeletons and cytosol. They are extraordinarily heterogeneous and their physical properties are strongly affected by the internally generated forces. In order to understand the out-ofequilibrium mechanics, we have developed a method of microrheology using laser interferometry and optical trapping technology. This method allowed us to probe mechanics and dynamics in living cells with a high spatio-temporal resolution. Microscopic probes in cells are stably trapped in the presence of vigorous cytoplasmic fluctuations, by employing smooth 3D feedback of a piezo-actuated sample stage. To interpret the data, we present a theory that adapts the fluctuation-dissipation theorem (FDT) to out-of-equilibrium systems. We discuss the interplay between material properties and non-thermal force fluctuations in the living cells that we quantify through the violations of the FDT.