I don’t really see where that comes from. They’re really small, but so are the Planck mass and charge and they’re not the smallest possible units of mass and charge. Rather obviously so, since the mass and charge of an electron is less than the Planck mass and charge.
Essentially, it’s because General Relativity describes the structure of space and time. We don’t yet have a quantum theory of GR, but we can still estimate the scales on which we expect it to become relevant by combining the constants associated with GR (G and c) with the constant governing quantum mechanics, h. The results are, of course, the Planck length and the Planck time.
Note, however, that it’s not entirely accurate to say that “the Planck length is the smallest possible length”. Rather, one should say that we have no reason to believe (or disbelieve) that our current models of space and time are applicable below this scale. It’s entirely possible that if and when we come up with a quantum theory of gravity, viewing spacetime as an infinitely divisible continuum will turn out to be valid; but it’s also possible that it won’t.
Oh, and as far as the Planck mass is concerned, you can view it as the mass of a black hole whose radius is the Planck length; so it’s related to the “minimum possible length” in that way, in the sense that we don’t know how smaller black holes would behave.
It’s also worth pointing out that most of the Planck scales are only estimates, based on dimensional analysis. Nobody would be particularly surprised if the fundamental quantum of spacetime (if there is such a thing) turned out to be 2pi times the Planck length, or 1/137 of it, or half of it, or the like.
The Planck action and the Planck speed are both fundamental limits, however, and are both known precisely. You can’t have an action (or change of angular momentum, which has the same units) smaller than the Planck action, and you can’t have a speed greater than the Planck speed. They’re better known as [del]h[/del] and c.