A cytoskeletal filament, with molecular motors walking on it, can be modeled as an elastic filament with a permanent distribution of stresslets along its contour. We show that an initially straight filament is unstable and can lead to autonomous filament motion which, depending on conformational symmetry, can be translational or rotational. Our results can be tested also on synthetic chains of autocatalytic particles or other linearly connected systems where chemical energy is converted to mechanical energy in a fluid environment.