Hey, if you ask me, any number is nonexistent in that it is merely an idea to express the quantity of a thing and, as it exists only as an idea, cannot be real. (Yes, even if you call it a 'real number.')
That said, you can divide by zero quite readily. What you do is you convert the numbers back from their abstract representations of reality back to the thing they represent, and then apply a bit of common sense to understand what zero means in a particular circumstance.
e.g.
You are traveling at exactly zero miles per hour. How long until you travel 300 miles?
300/0 = oh shi- oh right, I'm not moving, so I'll never get there.
You have 5 apples. You wish to split it between zero people. How many apples will there be per person?
5/0 = oh shi- oh, right, I'm not actually dividing the apples between anyone, so the question is irrelevant.