Is there a way to quantify the 'surface area' or simply the shape of organic compounds?
It is possible to compute the surface area of a molecules, and to compute the surface area of each individual atom in a molecule. These areas depend on the conformation of the molecule. An example of a software that does this is MSMS (http://mgltools.scripps.edu/packages/MSMS).

With respect to the shape, it is possible to categorize molecules in a triangular diagram describing similarity with "rods", "disks" and "spheres" (e.g. http://pubs.acs.org/doi/10.1021/ci300415d). Details about this analysis were published here: http://pubs.acs.org/doi/10.1021/ci025599w.