Space-time (ST) wave packets are diffraction-free, dispersion-free pulsed beams whose propagation invariance stems from correlations introduced into their spatio-temporal spectra. We demonstrate here experimentally and computationally that ST light sheets exhibit self-healing properties upon traversing obstacles in the form of opaque obstructions. The unscattered fraction of the wave packet retains the spatio-temporal correlations and, thus, propagation invariance is maintained. The scattered component does not satisfy the requisite correlation and thus, undergoes diffractive spreading. These results indicate the robustness of ST wave packets and their potential utility for deep illumination and imaging in scattering media, such as biological tissues.