What is a everyday example of a non measurable set?

Question

I'm working on my understanding of measurable sets and my immediate intuition wants to know what's not a measurable set? Initially I think of some space where divisions go to infinity, like a tolerance that just is defined to go on and on. For example plus or minus infinity from a to b would be unmeasurable. Does that even make sense? Furthermore measurable sets are topologies, which seemingly implies a very important property, because then one can find a defined function between any two measurable sets. In the end my biggest question is what's not a measurable set? Note: I'm learning measurability from the point of view a Lebesgue integral.