I'd like to apply a constant force to all objects, attracting them to the scene's origin. A Force object with negative strength at (0,0,0) with no falloff would do the trick.

Is it possible to disable a Force object's falloff?

For example, a negative force object will stretch a softbody; the closest parts of the mesh will move toward the force quicker than the rest. See this stretched torus object (attached .blend):