  We analyze the complex phenomenology of the Non-Classical Rotational Inertia
(NCRI) observed at low temperature in solid $^4$He within the context of a two
dimensional Berezinski-Kosterlitz-Thouless transition in a premelted $^4$He
film at the grain boundaries. We show that both the temperature and $^3$He
doping dependence of the NCRI fraction (NCRIF) can be ascribed to finite size
effects induced by the finite grain size. We give an estimate of the average
size of the grains which we argue to be limited by the isotopic $^3$He
impurities and we provide a simple power-law relation between the NCRIF and the
$^3$He concentration.
