I'll give this another spin:
Can music time signatures really be rational?
Which I'd answer: no, not really. Rationality is a mathematical concept, depending on an exact, axiomatic notion of numbers. Now, sure, any ordinary piece of music will use a rational signature given by integer numbers on paper, or in the DAW you use. Time is conceptually divided in a rational way in these compositions.
But is the conceptual level really what the music is? I wouldn't be sure. To the listener, music is first just fluctuations in the air pressure field. Our brains do a remarkable job of disassembling that mess again... they will notice that there are certain individual voices. They will perceive that these voices are somehow synchronised, that there are some scales of repetition etc.. If they know something about music they'll usually also be able to guess from that again what the numbers on the sheet were that the composer wrote.
But this is not quite an objective, reproducable process. Upon close investigation, you will find out that any human performance has slight tempo etc. fluctuations intrinsic in it. And, which may be more surprising, even a purely electronic, “exact” sequenced piece, when analysed from the audio mix, will seem to have such fluctuations, on a much smaller scale. The reason is actually mathematically akin to the quantum-mechanical Heisenberg uncertainty relation†: whenever you pack information into any sort of waves (be it electromagnetic radio waves or acoustic sound waves), you must make a tradeoff of frequency accuracy vs time accuracy. In exact terminology, transients in signal that has a frequency-bandwidth fU can have a time-accuracy of at most ≈ 1⁄fU.
Human hearing has a bandwidth of <20 kHz. Signals in that range have in principle a time certainty of at most 50 μs. In fact, humans can determine signal time only to about 3 ms... but whatever the exact numbers, there is in principle a limit to the precision. Therefore, it can never be possible to objectively distinguish a piece in 4⁄4 time from one in 4.00000001⁄4, or in fact from the irrational cosh(2.0634371)⁄4.
So why can we still be sure a given piece is in 4⁄4 time, and not in cosh(2.0634371)⁄4? Occam's razor. The simplest possible model that fits – within the available accuracy – the observed data, is the one that brings us closest to understanding what's going on.
Now there's an interesting detail: what do I mean by “the simplest possible model”? Simplicity can't really be defined, it is different for every human what seems simple and what seems complicated. In information theory, there exists a thing called Kolmogorov complexity. It's a tricky quantity, in fact it can't be computed – but it's still well-defined and can be decently approximated by the shortest program someone will send in to a given challenge on CodeGolf.StackExchange.
For example, 4⁄4 has a complexity of at most three characters or 24 bits, whereas the number 2.0634371 alone has a complexity of at least 30 bits.
But there are examples that are less clear-cut. In particular, there are some irrational numbers whose complexity we should consider rather low. The number π has a complexity of, like, one character, whereas the rational 3.1416 has already more than 16 bits complexity.
Therefore I would argue that a piece in π⁄4 should be considered to be in π⁄4 time, not in 3927⁄5000.
Can music time signatures really be irrational?
Yes they can, in every sense that the concept of time signatures can make sense at all.
†These uncertainty-effects actually have little to do with quantum fluctuations, physically – they're just mathematically analogous, but the physics is only classical mechanics.
Theoretically speaking, quantum mechanics does put an even more fundamental bound on how exactly we can measure time: time-uncertainty times energy-uncertainly must be larger than the Planck constant. (This can be seen as the reason why particle-physics experiments need to put in such enormous energies: some of the particles they're studying there have extremely short life-times, which can only be resolved by allowing for enormous energy-fluctuations.)
Applied to music, there is an energy-uncertainty of at least the average thermal energy that particles have at body temperature: Δ E ≥ k · Δ T = k · 37°C = k · 310 K ≈ 4.3×10-21 J. If you calculate the time-uncertainty from that, you get 1.5×10-13 s. Now, that's a time scale we can actually resolve with high-tech equipment (e.g. femtosecond lasers), way longer than the Planck time. But it's still many orders of magnitude smaller than the time-uncertainty due to classical effects, which I discussed above.