We know that, force between two charges \(q_1\) and \(q_2\) placed at distance \(r\) is given as
$\begin{aligned}
F & =\frac{1}{4 \pi \varepsilon_0} \cdot \frac{q_1 q_2}{r^2} \\
\Rightarrow \quad \varepsilon_0 & =\frac{q_1 q_2}{4 \pi F r^2}
\Rightarrow\left[\varepsilon_0\right]=\frac{\left[q_1\right]\left[q_2\right]}{[F]\left[r^2\right]} \\
& =\frac{[\mathrm{AT}][\mathrm{AT}]}{\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{L}^2\right]}=\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^4 \mathrm{~A}^2\right]
\end{aligned}$