I'd thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable

Well, the number of points in a Hilbert space of dimension 2 is uncountable, and yet the space has dimension 2!

I suspect the source of the confusion here is that you're trying to think of the values of a function as its "coordinates". But this is wrong: the "coordinates" are the coefficients of a Fourier series expansion of the function.

The confusion is understandable, given that the two concepts coincide in the finite-dimensional case. You should think of a point in C^2, say (3,7), as a function from the two-element set {1,2} into the complex numbers C (in this case we have f(0) = 3 and f(1)=7). You can then write every such function uniquely as the sum of two "basis" functions: one that sends 1 to 1 and 2 to 0 (call this b_1), and one that sends 1 to 0 and 2 to 1 (call this b_2). Thus f = 3b_1 + 7b_2, i.e. (3,7) = 3(1,0) + 7(0,1).

In the case of functions on the real line, however, the "basis" functions cannot be functions that send one number to 1 and the rest to 0, because in order to represent an arbitrary function, you would need to add uncountably many such things together, which is not a defined operation (or, more technically, is only defined when all but countably many of the summands are zero).

Fortunately, however, if we're talking about the space of square-integrable functions (and that was what we wanted anyway, wasn't it?), we do have a countable orthogonal basis available, as was discovered by Fourier.

(Technical note: Actually, the space of square-integrable functions on (an interval of) the real line doesn't consist of well-defined functions per se, but only of equivalence classes of functions that agree everywhere except on a set of measure zero. See measure theory, Lebesgue integration, L^p space, etc.)