Solution based measurement of Solvent-Accessible Surface Area of macromolecules
As you noted, the solvent-accessible surface area is a geometric concept, and as such is most easily calculated using computational geometry tools from a given molecular (or crystalline) structure. However you obtain this atomistic structure (X-ray, neutrons, molecular simulation, NMR, …) is of course of significant importance to the resulting area, but the calculation method will be the same.

There are, however, a few studies which have attempted to link the solvent-accessible surface area to directly measurable properties, and in particular to spectroscopic properties. A quick search of the literature reveals three examples:


Amide H/2H Exchange MALDI-TOF Mass Spectrometry
UV Raman directly on the exposed residues
Using diazirine as a probe