I'm confused over the size of these extra dimensions... They say that they do not rule out small compact extra dimensions, but only large ones, the number they give is ~100 km. But this must surely be ruled out by ordinary gravitational experiments that establish the 1/r^2 law for gravity, since I would assume that that law would be seriously broken for extra dimensions of those sizes for even everyday physics.

The number that is quoted in some textbooks on String theory (but I do not have the refs.) is that gravity is 1/r^2 down to sizes of ~1 cm for extra dimensions.