Some of the values of $1-P_i$ may be impossible to estimate by hand, which leaves us with the possibility of estimating them by random sampling on a computer. But the definition of $M_i$ may not be specific enough to allow software to determine whether it is "the" reason that a certain position is unreachable. Also, $1-P_i$ may be too small to make it possible to find any reachable positions in a random sample. And determining the reachability of a position may take too long -- it may itself be an intractable problem in general.

Some of the $P_i$ may be impossible to estimate by hand as Joel David Hamkins did for $P_1$, which leaves us with the possibility of estimating them by random sampling on a computer. But the definition of $M_i$ may not be specific enough to allow software to determine whether it is "the" reason that a certain position is unreachable. Also, $1-P_i$ may be too small to make it possible to find any reachable positions in a random sample. Or even if $1-P_i$ is 0.5, we may be unable to demonstrate that by sampling, because determining the reachability of a single position may be an intractable problem in many cases.

Here is an argument that suggests that estimating this number, even to within many orders of magnitude, is intractable.

As an illustration, consider two sets of positions. A is the set of all positions in which white has 8 pawns, 2 bishops, and no queens, and black has the same. B is the set of all positions in which white has no pawns, 5 bishops and 5 queens, and the same for black. B is about 30 times bigger than A. We should expect that most positions have this character: boards crowded with powerful pieces as a result of many pawn promotions, including a lot of underpromotions.

And B is only one of many similarly defined categories. Each of these lives on some huge brushy branch of a game tree, which may have many disconnected parts and whose parts may or may not be connected to the starting position of chess so that they're legally reachable. Each category or subcategory may require significant theorem-proving in order to determine its contribution to the total number of reachable positions. The total amount of analysis to be completed may be many orders of magnitude greater than the amount of analysis that has ever actually been done on practical chess.

This answer was originally a specific argument that the problem might be intractable due to dominance by positions that are unreachable for a specific reason. I've rewritten the answer to be more general.

Joel David Hamkins' answer has put an upper bound on the result. The bound comes from a certain mechanism, a specific constraint involving the arrangement of the pawns. Let's call call this mechanism $M_1$.

Let $x$ be the fraction of reachable positions. Suppose our goal is to put bounds on it, $a<\log_{10} x < b$, with a relatively small value of $\Delta=|a-b|$. Mechanism $M_1$ gives $b=-9.4$. Douglas Zare's answer estimates $10^{47}$ positions, and if, say, at least $10^6$ distinct positions have been reached in real games, we have $a=-41$. That gives us $\Delta\approx 32$, which is pretty wide. I would consider the problem intractable if this can't be improved to something more like $\Delta=4$.

Here is a second mechanism, $M_2$, which may also make many positions unreachable. As an illustration, consider two sets of positions. A is the set of all positions in which white has 8 pawns, 2 bishops, and no queens, and black has the same. B is the set of all positions in which white has no pawns, 5 bishops and 5 queens, and the same for black. B is about 30 times bigger than A. We should expect that most positions have this character: boards crowded with powerful pieces as a result of many pawn promotions, including a lot of underpromotions.

Some folks have expressed skepticism in comments that $M_2$ really makes very many positions unreachable. I don't know -- all I've offered is a plausibility argument. The question arises of how one would ever establish the answer reliably and verifiably. I don't think it helps much to construct and analyze sample positions as suggested in Douglas Zare's comment, because this proves nothing about the probability in general that a position is unreachable due to $M_2$. Possibly some kind of random sampling would work.

The answers so far seem to have focused on looking for insight into mechanisms $M_i$ that prevent a position from being reachable, and then trying to estimate the probability $P_i$ that a randomly chosen position is unreachable due to that mechanism. We could then guess $\log x=\Sigma \log (1-P_i)$, assuming that the probabilities are independent. But there are some real problems with this approach.

First and most importantly, we can't necessarily enumerate all the mechanisms $M_i$ or convince other people that we've enumerated them all.

Some of the values of $1-P_i$ may be impossible to estimate by hand, which leaves us with the possibility of estimating them by random sampling on a computer. But the definition of $M_i$ may not be specific enough to allow software to determine whether it is "the" reason that a certain position is unreachable. Also, $1-P_i$ may be too small to make it possible to find any reachable positions in a random sample. And determining the reachability of a position may take too long -- it may itself be an intractable problem in general.