At a high school level, whether an equilibrium constant has units is a function of the actual equilibrium.
If the number of moles of product and reactant are the same, such as in the generic alcohol + acid <-> ester + water, then Kc will not have any units, but something like BaseH+(aq) <-> Base(aq) + H+(aq) will have units.
Similarly, a reaction like H2 + Cl2 <-> 2HCl will not have units for Kp, but CaCO3 <-> CaO + CO2 will.
(double headed arrow meant to represent the equilibrium arrow)
I assume, from the area you've posted the topic in, that this is the more appropriate situation and sort of answer you're looking for.
At higher levels you learn about activity coefficients and reaction quotients, which are factors multiplied to concentrations or similar to allow for effective concentrations or pressures (a typical example being in concentrated solutions of strong acid, dissociation is not complete, so pH != -log[H+]), these too have the effect of giving no units for any equilibrium constant, K.
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