Right, and I'm sure you're aware of the Planck constant for energy quanta as well. That's the kind of thing that made me say that a base unit of length is "plausible".
So although it's a fun thought experiment to think about strings that that are precisely 4998997308233 base units long, actually measuring such a thing is too expensive or even impossible due to Heisenberg.
Nevertheless, mathematics has demonstrated that it's useful to think of "real numbers" as infinitely precise things, because that way your number system doesn't impose any preordained limits on your measurements -- even though nature itself does.
By far most real numbers are actually non-computable, meaning that there is no finitely expressible procedure for listing their digits to any desired length.
Now it's always seemed clear to me that if a thing is fundamentally unobservable and unidentifiable in any way, you might as well say that thing does not exist at all. Nevertheless, the theory of real numbers implies that the uncomputable numbers "exist" in some sense.
That actually simplifies the theory. Otherwise you'd have to confine yourself to the computable numbers, namely all strings of binary digits that can be produced by some Fexl function (see http://fexl.com/). For example, the number .1010... could be expressed as:
\number == (1; 0; number)
(I use Fexl because it's based on combinatorics, which behave according to very simple rules. Ultimately any Fexl function can be expressed as a binary tree with only "S" and "C" at the leaves.)
That might make the strict constructivists happy, but it might also hamper the free reigning thought processes of mathematicians.
There is no discrete base unit of space, and on top of that objects do not have determinate sizes.