What is an "un-reproducible result"? In order to publish a research paper in a journal, one of the requirement may be the reproducibility of results.
I'm wondering, in academia, how does an un-reproducible result even exist?
Is there any classical example for it, in any branch of academia research?
It's important to define what reproducibility actually means and in what context it is used. Science deals with things that can be reproduced in principle: If you managed to re-create the exact same situation, you would be able to get the same result.
But in practice, that doesn't always mean that you can re-create the situation: You might have measured the seismic waves of a very large earthquake in Indonesia. Or you might have seen photons of a nearby supernova. Neither of these conditions can be created by humans, and so the experiment can not be repeated in practice, though in principle it could be. A related situation happens if it is impractical to do so: If the original experiment was done with a ten-billion $ machine (say, a particle accelerator, a nuclear fusion reaction), then yes you could repeat the experiment, but you probably find yourself in financial trouble if you tried. There are also valid research results that should not be reproduced, even if they could: Say, whatever we may have learned from the Tuskeegee syphilis study or the Stanford prison experiment might be scientifically correct, even repeatable, but one can only hope that nobody will ever try to repeat these studies.
Finally, there are often practical constraints: If you take a picture of turbulence in a pipe, you will not be able to recreate the same picture because turbulence is a chaotic process; similarly, if you try to do experiments on a single cell and count the number of molecules of a specific kind, you're likely going to find that it depends sensitively on temperature, time of day, etc. That doesn't mean that the science is wrong: In both cases, statistical assessments of the results may still be valid, even if you can't recreate the specific numbers.
Of course, there are also experiments that really can't be reproduced: Someone published the results of an experiment that seemed reasonable to them and to the reviewers, but the measuring device had a mechanical defect and consequently every number in the publication is just wrong and the measured effect does not actually exist. This of course shouldn't happen, but it does happen in practice. There are also common statistical problems in studies that involve a small number of human subjects where the random, involuntary, or voluntary choice of subjects suggested an effect that, if repeated on a larger and more random cohort does not actually exist.