Coulomb's law in vacuum is generally stated as

$$F = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r^2}.$$ However, apparently, by going to Gaussian units, $\varepsilon_0$ will take on a value $\frac{1}{4\pi}$, thus the law will be reduced to

$$F = \frac{Q_1 Q_2}{r^2}.$$ I can't wrap my head around this! On the two sides of the equation, there are quantities of different dimensions! Even if the numerical values are equal, it feels like we are throwing dimensional analysis away.