What is the unit for resonant frequency?
In the simplest way possible:

L is in henries (H) - \$\Omega \cdot s\$.

C is in farads (F) - \$\dfrac{s}{\Omega}\$

Multiply both and you have \$s^2\$. Take the square root, you have \$s\$. Invert it, you have \$\dfrac{1}{s}\$, that is, \$\frac{rad}{s}\$.

If the expression is written as \$\omega_0 = \dfrac{1}{2\pi\sqrt{LC}}\$ the resonant frequency is in hertz.