In the Ancient Greek world (and even during the Roman Republic), theatre often occured as part of civic festivals and celebrations. So, few if any people would be working on the days when the theatre was active (see Wiles 2000, Green 1996, and Arnott 1991). But your question about the weather is fantastic and, luckily, I found something dealing with EXACTLY that issue. In their article, Chronopoulou and Mavrakis analyze "the writings of the dramas of Aeschylus, Sophocles, Euripides and the comedies of Aristophanes during the Golden Age...These theatrical texts provide descriptions of clear and stable weather conditions." Their major conclusion is that "The comedies of Aristophanes, especially those presented in Lenaia, often invoke the presence of the Halcyon days. Combining the fact that dramatic contests were held in mid-winter without any indication of postponement, and references from the dramas about the clear weather and mild winters in Attica, we can assume that those particular days of almost every January were summery in the fifth and maybe in the fourth centuries BC. Information is also drawn from the paintings on vessels showing that the clothes worn in Lenaia and in the wedding ceremonies were not designed for rainy weather. All these references concern indications for the fifth century BC. We found no weather indications in the fourth century BC dramas, but dramatic contests continued to take place during that century, because Aristotle’s references to Lenaia were in the fourth century BC."

References:

Christina Chronopoulou and A. Mavrakis. "Ancient Greek drama as an eyewitness of a specific meteorological phenomenon: indication of stability of the Halcyon days." Weather Vol. 69, No. 3 (March 2014): 66-69.

David Wiles. Greek Theatre in Performance: An Introduction. Cambridge: Cambridge University Press, 2000.

J. R. Green. Theatre in Ancient Greek Society. London: Routledge, 1996.

Peter D. Arnott. Public and Performance in the Greek Theatre. London: Routledge, 1991.