When measuring a physical length (or anything else corresponding to the position of a pointer on a scale) we do so by first counting how many copies of a basic “unit” length fit into the interval. This gives an upper and lower bound on the length (of the form n<L<n+1). If we want a more accurate estimate we use some fraction of the unit (tenths in the decimal system, but also possibly halves, thirds, twelfths and so on). Repeating the process gives us a sequence of successively smaller intervals in which we can say that L lies, and although we always stop at some point we have to acknowledge that this leaves us with a non-zero error bound. The true value of L corresponds in principle to a sequence of rational approximations which it seems can be made as close as we like if we work hard enough. This is exactly the way “real” numbers are defined, so one reason for calling them the “real” numbers is because they correspond to the (practically unattainable) “true” values of real measurements.
