I mean that for any epsilon > 0, you can have a set of intervals of the form (a_i, b_i) where every rational number is in some interval and the sum over all i of b_i - a_i < epsilon.

That is, you can cover the rationals with intervals of arbitrarily small total length.

I see. But is that not just a simple conclusion of the fact that the rationals are countable, and that there exists a converging series of positive numbers?

Right but explaining why the reals are different is the tricky part