We present a resonant inelastic x-ray scattering (RIXS) study of spin and charge excitations in overdoped La1.77Sr0.23CuO4 along two high-symmetry directions. The line shape of these excitations is analyzed and they are shown to be highly overdamped. Their spectral weight and damping are found to be strongly momentum dependent. Qualitative agreement between these observations and a calculated random-phase approximation susceptibility is obtained for this overdoped compound, implying that a significant contribution to the RIXS signal stems from a continuum of charge excitations. Furthermore, this suggests that the spin excitations in the overdoped regime can be captured qualitatively by an itinerant picture. Our calculations also predict a low-energy spin-excitation branch to exist along the nodal direction near the zone center. With the energy resolution of the present experiment, this branch is not resolvable, but we show that the next generation of high-resolution spectrometers will be able to test this prediction.