In my collection, I have this 1940s press photo, given out as part of a pack of all American Association ballparks to radio announcers, which gives us the basic field layout:
It lists the dimensions as 267 feet down the left field line, 268 down the right field line and 392 feet to straight away center.
On this thread on Baseball Fever, user "Pelt" created worked out the rest of the layout in sketchup:
He calculated a distance of 435 feet to each of the corners, "with only a couple of inches difference between the two".
That's an awful lot of real estate for an outfielder to cover.