Can multi-rate operations of decimation and interpolation can be used to implement a rate change by a factor an irrational number?
You're exactly right, with rational operations, you can't achieve an irrational ratio.
The only way out there is using some interpolative number, i.e. an arbitrary resampler.
However, in signal processing, you basically always deal with system that only run for a finite time and within that have a boundable tolerance against jitter or frequency error - so one common solution to that dilemma is, e.g. in receivers, to do the resampling with a slightly wrong, but rational ratio, and then let the symbol synchronizer, which often does some form of arbitrary resampling internally, anyways, take care of the rest.