The product of two irrational numbers will be either  a rational or an irrational number.
Consider the following example: $(\sqrt{3}+\sqrt{2})\times (\sqrt{3}-\sqrt{2})=1$. 
Both $(\sqrt{3}+\sqrt{2})$ and $(\sqrt{3}-\sqrt{2})$ are irrational numbers and their product is 1. 
1 is an integer and since, all integers are rational numbers as well, we can infer that product of two irrational numbers may be an integer, a rational number or an irrational number and they are all real numbers.
When we multiply the irrational numbers $\sqrt{2}$ and $\sqrt{3}$, we get $\sqrt{6}$, which is also an irrational number.