What is the most physically accurate model of surface materials possible in computer graphics? I  am looking for theoretical and implemented models to represent surface materials in software that strives for physical accuracy.

Is BRDF really the best model? What alternatives are better? Why? What drawbacks beside performance do these models have?
BRDFs are not accurate for any dielectric surface, as all of them have some degree of subsurface scattering. A BSSRDF is also just an approximation based on the assumption of a semi-infinite slab. Ideally, you'd run a volumetric path trace for all surfaces, which would be much more accurate than BRDFs or BSSRDFs, but significantly more expensive. Although, Disney has already used brute force path tracing over BSSRDFs in production, see the SIGGRAPH 2015 course notes, section 2.5.