[logicchains]

1/3 is a rational number, so intuitionism is fine with it. It's also fine with "computable reals": real numbers that have an algorithm for computing them (that algorithm can be represented with a finite amount of information). What cannot be constructed in intuitionism is a "non-computable real": a real number for which there is no algorithm to compute it. The vast majority of classical real numbers are non-computable.

Another way to look at it: if we have a limiting process that approaches some number X, classically we could use the law of the excluded middle to prove that it must eventually reach X, hence X "exists". Under intuitionism however we cannot say that X "exists" or construct X, all we can do is construct the process that approaches X and say this process exists.