Maybe "connectedness" is the notion you're trying to get at -- the real numbers are topologically connected, but the rationals aren't. If "A" is the set of rational numbers x with x^2 < 2, and B is the set with x^2 > 2, then the rationals are the union of A and B, and there is a "hole where sqrt(2) should be", so the rationals are disconnected. It's possible to define the word "connected" in a way that makes this notion precise.
A related notion is what's called "(sequential) completeness". The infinite sequence whose terms are (2, 2 + 1/2, 2 + 1/2 + 1/6, 2 + 1/2 + 1/6 + 1/24, ...), where the nth term is obtained by adding 1/(n!) to the previous term, intuitively "should" converge, since its elements get arbitrarily close together as n gets arbitrarily large. Any such sequence converges to a real value (this one converges to the exponential constant "e"). But if our number system is only countably infinite, there must be some sequences that get arbitrarily close together but don't converge. For example, if we restrict ourselves to rational numbers, this is a valid infinite sequence (every element is rational), and its terms get arbitrarily close together as "n" is large, but it doesn't converge to anything.
A profound question to reflect on; and interesting to contrast with modern science as a philosophical foundation that there might not be any continuous objects present in reality. Might be discrete all the way down.
In a sense, it seems like the uncomputable reals are an artifact of assuming continuity, ie, between any two numbers located on a line there must be more numbers. Part of the reason it is so unintuitive is we don't have any real lines to play with at the physical human scale, they fall apart at the atomic level and turn out to be non-continuous approximations.