If a photon (and graviton, electron, etc.) had 4 degrees of freedom, our world would be 4-dimensional (obeying inverse cube law). If a graviton were 5-dimensional on scale, say, less than 1 mm and 3-dimensional on bigger scale, then below 1 mm gravity would obey 1/r^4 law but above 1/r^2.

So on a large scale we live in 3-D world and do not have more degrees of freedom to move. We just made out of such objects which only have 3 large degrees of motion (electrons, protons, photons, etc).

Yet they can have many small degrees of freedon to move (we call the result of motion in those degrees as a spin, charge, strong charge, etc).