Even 'women's work' can become a male preserve when it is elevated to haute couture or haute cuisine , when it is no longer a necessity but an art. Athenian women may have been excluded from even this entree into the male-dominated world of theatrical performance. A peplos for Athena may have been considered very different from a costume for Dionysus. But whoever ran it and labored for it, an industry there must have been. If an individual garment could take months to make, there would have to be a great many weavers at work to keep up with the demand.
Roman theater is a slightly different question, but one worth asking.
The fabulae palliatae of Plautus and Terence may have required only simple tunics and cloaks of the sort which could be bought easily. The very name of the fabulae togatae indicates how the performers were dressed. Nothing off the rack, I'm sure. Professor Norma Goldman of Wayne State University in Detroit, Michigan, has been extensively involved in re-creating Roman fashion as part of a project at the American Academy in Rome. There are plenty of vases illustrating phlyakes comedies in southern Italy and numerous figurines, etc., of Roman actors.