The paradox is clearly not about barbers or sets.
It can feature any verb or verb phrase - shave, adore, paint, include in a set, share a pizza with - that can take a subject, object, and crucially, a tense. Almost any verb you like in fact. What we're concerned with here however is the logical tense. This kind of tense isn't necessarily overtly marked by any grammatical suffix like "-ed" in English, or by a word like "then" or "now" or "tomorrow". But it is demonstrated just by the repetition of the verb in a compound proposition or sequence of such propositions. Each constituent clause or proposition featuring the same verb, even with the same grammatical tense, has a different logical tense and refers to a different time.
An artist paints all and only those who don't paint themselves. Each of the two occurrences of "paint" have a different logical tense since they occur at different place-times in this sequence of clauses. We can make this difference highly explicit: An artist will paint tomorrow all and only those who didn't paint themselves yesterday. So if the artist didn't paint herself, then she will. And if she did, then she won't. Simple. Logical. The set will include all and only those sets it didn't include before, so if it did include itself before then it won't next time, and vice versa. Any paradoxical contradiction is merely a result of failure to differentiate with respect to logical tense, of flattening out the entire argument into a kind of timeless present.
This shouldn't be an entirely unfamiliar idea. We use brackets to indicate arithmetical tense, that is the order in which operations are to be performed, to avoid contradiction.
Time heals all logical wounds.