I will suppose that with "rational with respect to a base" you mean that the "decimal" expansion in this base is finite or periodic and with "irrational" you mean the complement.
You can simply choose a transcendental number as base:
http://en.wikipedia.org/wiki/Non-integer_representation
Any periodic or finite representation will lead to a fraction involving $e$, but they are all irrational (in the ordinary sense), so a rational number (in the ordinary sense) between 0 and 1 cannot represented that way.
Note that irrational bases are somewhat tricky, the wikipedia link has some information on this.