Answer
True.
Work Step by Step
Suppose there exists some irrational number such that its square root is rational. Then, the square root can be expressed as $\frac{a}{b}$ for positive integers $a$ and $b$. Then, the number is $\frac{a^2}{b^2}$. However, both the numerator and denominator of this number are integers, as they are each the square of an integer, so the number must be rational, but this is a contradiction, so the statement is true.