I heard that one of Euler's Beta/Gamma function identities models the strong force. I was just wondering how it did this. (This might be a stupid question) How do we measure the strong force, and how is it a function of two variables?
The ##\beta## function is a renormalization-group quantity describing how the coupling constant "runs" with the change of the momentum-renormalization scale. It has nothing to do with Euler's beta and gamma function, except that these occur in a natural way when using dimensional regularization, which is common practice in perturbative QCD and all modern practitioners of relativistic QFT because it's a very convenient regularization scheme for gauge theories.
I suspect the OP is referring to the Veneziano amplitude of open-string theory and the Virasoro-Shapiro amplitude of closed-string theory, which model high-energy hadron scattering, and which involve the Euler beta and gamma functions: http://en.wikipedia.org/wiki/Veneziano_amplitude