image based example for Bayes error/irreducible error
As an MNIST dataset, see the mispredicted images on this figure below,  you notice for some samples, we as human don't know for sure what is the true label.



Maybe there are other factors that are not visible to us to predict the correct output. 

Assume I want to predict customer will leave our company, maybe there are tons of useful features to start with, yet he may simply have some other problems that we are not aware of, and it may affect his decision in leaving the company.