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.