All speaking parts in both Classical Athenian tragedy and comedy were played exclusively by men, or at least boys. This includes both the actual actors and the chorus. While women might appear in comedy and lower forms of theater like mime and pantomime they did not speak--their role in comedy was generally basically as a stripper. They did not appear in tragedy at all. This isn't really all that unusual--Japanese Noh plays did not feature female performers, and even early opera used castrati to play the female roles, although in the case of opera I don't know if that was exclusive. The use of masks, as in Japanese Noh, and the costumes would've rendered the actual sex of the performer pretty irrelevant to the audience--you can get a sense of that through this modern English performance of the Agamemnon, which is quite good (especially in relation to its choruses), or through watching the masked performers of any Noh play

/u/XenophonTheAthenian has answered the question, but to expand on this we aren't even sure whether women were allowed to watch classical Athenian tragedy. There are some ancient texts that imply they were permitted at the theatre, for example Plato, who writes in Laws that actors should not be permitted to

harangue women and children and the whole populace^1

or in Gorgias, where Socrates says that tragedy is

a kind of rhetoric addressed to such a public as is compounded of children and women and men, and slaves as well as free^2

(The sceptics' counter-argument is that neither of these quotes are talking about the actual performance context of Greek drama: the Athenian Stranger in Laws is clearly discussing a counterfactual, while Socrates in Gorgias is dismissing tragedy as mere demagoguery that pleases children and women, rather than philosophy which has a serious point.)

There is also a joke in Aristophanes' play Peace which, it has been argued,^3 makes more sense if we assume that women were permitted to attend:

Trygaeus: Wait. I take this fire-brand first and plunge it into the water. Now quick, quick, you sprinkle the altar. Give me some barley-seed, purify yourself and hand me the basin; then scatter the rest of the barley among the audience.

Servant: Done.

T: You have thrown it?

S: Yes, by Hermes! and all the spectators have had their share.

T: At least the women got none.

S: Oh! their husbands will give them some this evening.^4

The joke is on the Greek word for "barley-seed", which is also a slang word for "penis"; Henderson argues that the joke makes more sense if the women are in the audience,^5 but e.g. Simon Goldhill is more sceptical and points out that the joke still does make sense if women didn't attend.^6

Finally, there are some later anecdotes which mention women in the theatre: e.g. the Life of Aeschylus (an anonymous and post-Classical biography) mentions that watching the Eumenides frightened some women so much that they had miscarriages. Those who argue that women were permitted to watch classical theatre claim that these later anecdotes might have been preserved from fifth century sources and thus shouldn't be dismissed; those sceptical of the claim point out that we have no idea where the tradition came from and if it does have any fifth-century origins.

Basically everyone agrees that it's unlikely that the question will ever be solved to everyone's satisfaction, and it essentially comes down to whether you think that the religious or civic function of classical Athenian theatre was more important.^7 (As women were certainly permitted to attend religious festivals such as the Panatheneia, and certainly weren't permitted to take part in political and legal life).

[Translations taken from Perseus Digital Library versions]

1. Plato, Laws, 817c

2. Plato, Gorgias, 502d

3. By e.g. Jeffrey Henderson, "Women and Athenian Dramatic Festivals". 1991.

4. Aristophanes, Peace, 955-70

5. Henderson's thesis is that women were allowed in the theatre, but had to sit in a separate section to the men. Thus, in his view the joke is that the slave throws barley to all of the sections except for the one the women were in (either comically avoiding the women's section, or because the women's section was at the back and he couldn't reach that far), as well as the pun.

6. Goldhill, "The Audience of Athenian Tragedy" in The Cambridge Companion to Greek Tragedy. 1997.

7. The most recent discussion that I know of, which explicitly makes this point, is a chapter in David Kawalko Roselli, Theater of the People: Spectators and Society in Ancient Athens. 2011.