There is no discrete base unit of space, and on top of that objects do not have determinate sizes.
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)
That might make the strict constructivists happy, but it might also hamper the free reigning thought processes of mathematicians.
A physical object does not have a sharp edge.
Now presumably 1 meter will be equivalent to some integer multiple of base units, though that may turn out not to be the case. Perhaps the current definition of a meter in terms of certain wavelengths of light will turn out to have a remainder of 1/7 of a base unit. ;) In that case you'd just define a slightly adjusted meter as a whole number of base units. So for simplicity let's assume that a meter is a whole number of base units.
Then it's clear that a string of any integer meter length would be possible. A string of length 1/3 meters might be possible, assuming that the number of base units in a meter was divisible by 3. But a string of length pi meters would be impossible.
As a computer programmer I tend to think of transcendental numbers in terms of processes which relentlessly converge, and that of course is the theory of limits. But I also recognize the financial constraints on running processes. So there will always be a "good enough" aspect to any physical measurement.