If the idea is to come up with a rough figure, the following
approach using labeled pieces might help.

Consider the white king's pawn.  It cannot inhabit 24 of the
squares on the chessboard  (I assume promotion is strict
and monotone.  I let others argue whether to add 8 to the
number 24.) .  For the other king side pawns 24 grows to 26, 30,
and 36.  Starting with these ratios, a set of four pawns labeled
with their starting squares can occupy about 9% of all possible
positions available.  Raise this to the fourth power, and one
gets a ratio of 6x10^-5 as an upper bound on positions of
labeled pawns, where as a rough count I include pawns sharing
the same square.

Now the bishops provide another factor of about 6x10^-2,
the two kings together a factor of about 7/8, and the rest provide a factor
that is likely closer to 1.  This gives a weak upper bound for
all 32 labeled pieces of 3 x 10^-7 fraction of legal positions to all positions.

I imagine with refinements one can probably subtract one more
from the exponent.  Hopefully the fraction involving unlabeled pieces will not be much different.

I just noticed domotorp's comment regarding positions having just 28 pieces.  While this suggests the fraction
of legal positions may be higher, for the labeled case, the ratio (all possible for 28) to 
(all for 32) should still be small, and should have a small effect on the rough calculation.