The WKB approximation states that in one dimension, the tunneling probability $P$ can be approximated as
$\ln P=-\frac{2\sqrt{2m}}{\hbar}\int_a^b \sqrt{V-E} dx$ ,
where the limits of integration $a$ and $b$ are the classical turning points, $m$ is the reduced mass, the electrical potential $V$ is a function of $x$, and $E$ is the total energy. Setting $V=kq_1q_2/x$, we have for the integral
$I=\int_a^b \sqrt{V-E} dx$
$=\frac{kq_1k_2}{\sqrt{E}}\int_{A}^1\sqrt{u^{-1}-1} du$ ,
where $A=a/b$. The indefinite integral equals $-u\sqrt{u^{-1}-1}+\tan^{-1}\sqrt{u^{-1}-1}$, and for $A\ll 1$ the definite integral is then $\pi/2$. The result is
$\ln P=-\frac{\pi kq_1q_2}{\hbar}\sqrt{\frac{2m}{E}} $ .
This result was obtained in Gamow 1938, and $G=-(1/2)\ln P$ is referred to as the Gamow factor or Gamow-Sommerfeld factor.
The fact that the integral $\int_A^1\ldots$ can be approximated as $\int_0^1\ldots$ tells us that the right-hand tail of the barrier dominates, i.e., it is hard for the nuclei to travel through the very long stretch of $\sim 1$ nm over which the motion is only mildly classically forbidden, but if they can do that, it's relatively easy for them to penetrate the highly classically forbidden region at $x\sim1$ fm. Surprisingly, the result can be written in a form that depends only on $m$ and $E$, but not on $a$, i.e., we don't even have to know the range of the strong nuclear force in order to calculate the result.
The generic WKB expression depends on $E$ through an expression of the form $V-E$, which might have led us to believe that with a 1 MeV barrier, it would make little difference whether $E$ was 1 eV or 1 keV, and fusion would be just as likely in trees and houses as in the sun. But because the tunneling probability is dominated by the tail of the barrier, not its peak, the final result ends up depending on $1/\sqrt{E}$.
Because $P$ increases extremely rapidly as a function of $E$, fusion is dominated by nuclei whose energies lie in the tails of the Maxwellian distribution. There is a narrow range of energies, known as the Gamow window, in which the product of $P$ and the Maxwell distribution is large enough to contibute significantly to the rate of fusion.
Gamow and Teller, Phys. Rev. 53 (1938) 608
Best Answer
They are mixing two different things here.
The strong force does not work between protons and neutrons, it works between quarks. As a side-effect of the way it works, it also constantly creates new particles, mesons. This particle-creation process is conservative in that if you consider all of the particles that are created, their momentum, charge, spin and so on all add up to zero.
It's all those mesons interacting with each other and the quarks that give rise to a second force, the nuclear force. It is the nuclear force that "acts on protons and neutrons to keep them bound to each other inside nuclei", not the strong force. Of course, one is the result of the other, so half full sort of thing... and that's why you see it called the strong nuclear force, or the strong interaction or all sorts of other names just to confuse things.
The distance that the nuclear force operates over is simply a function of the mass of the mesons and the uncertainty principle; all virtual particles with mass have a maximum lifetime, and if you simply see how far these mesons can go in that time, presto, you get a distance.
UPDATE: I realized I missed the closure.
The weak force is also mediated by massive particles, the W's and Z's. Like the mesons in the nuclear force, they are thus subject to the same range limitations due to the uncertainty principle. However, the mass of a simple meson like a pion is about 100 MeV, while the Z is a whopping 90 GeV. That's heavier than an entire iron nucleus! Now it might sound odd that such a heavy object can be created ex nihlo inside something like a helium nucleus, which is way lighter, but that's the whole idea of the uncertainty principle, for a very short time this is allowed, and that's why it's range is so short and the reaction is so rare in comparison.