I am told that the theories of General Relativity and Quantum Mechanics are fundamentally incompatible... Why is that?
Someone explained that it had to do with the fact that quantum particles such As quarks and photons are points without volume... Leading to singularities.
Do people really believe that ^
What data do we have to suggest that quarks are really zero dimensional points?
You're on the right track. However, it's not GR that's incompatible with singularities (Black Holes are an example of singularities in GR): it's QM. Under quantum mechanics, particles are not treated like singularities. They're instead characterized by wave functions, which introduces uncertainty in their position, momentum, etc. and quantizes properties like their energy. The real reason we haven't been able to reconcile QM with GR is that in order to so, we'd have to quantize the gravitational field; i.e. break it up into a countable series of discrete states that it can be excited into, which is what QM does with all other fields. And the problem is that when you try to do that, you end up with a condition called non-renormalizability, which essentially means you have infinities all over the place that can't really be dealt with unless you have some parameters from another source (say, string theory).
