There is rule for a number to belong to the set of real numbers.
It's called 'Law of Tricotomy'. According to it, "Any two numbers a, b are real if and only if one of the following holds true (i)ab.
In short a number is real if and only if it can be compared (like lesser than or greater than or is equal to).
As for the question 0 can be compared, right?
Hence, 0 is a real number.