"Ancient Greek society" was not a constant; it existed over a long period of time. It is clear to anyone comparing the Iliad to the writings of Aristotle that the position and stereotypical image of women changed a lot over the course of that period - and not for the better.

In early Greek writings, women were treasured for their skills and their counsel. A good wife was part of what gave a man honour and status in the eyes of others, and such wives were not hidden away or silenced. Meanwhile, their monopoly on the skill of weaving, which at this time was entirely a cottage industry, meant that they provided a significant chunk of household wealth.

Over time, however, marriage became much less about shacking up with a good wife, and much more about making the right alliances. Pedigree and connections were everything to the elite families of late Archaic and Classical Greece, and as a result everything was done to ensure that the right marriage ties were made and that a wife produced offspring, which was now her main task. Upper-class women of this time had no work, no political rights, no public representation; they could not own property; ideally, they were not seen outside the house. The fifth-century BC Athenian politician Perikles famously said that the best kind of wife was one about which no one spoke, whether in praise or reproach.

Source for the above is mainly Hans van Wees' article The invention of the female mind: women, property and gender ideology in Archaic Greece, which you can download through chs.harvard.edu/CHS/article/displayPdf/386 - I recommend giving it a look!

A couple of points in addition. First, as Xenophon makes clear in his Oikonomikos ("on household management"), elite women had very significant responsibilities in the household - controlling slaves, managing stock and finances, overseeing the budget and raising children. This allowed the husband to spend his days in the field, in the market, or in politics. It was therefore obviously a very valuable thing to marry an intelligent and competent woman, rather than just a trophy wife.

Second, what I've sketched above was the ideal among the elite. The normal population would not have been able to afford keeping their wives indoors and inactive. We know that lower-class women would have had jobs, gone out to buy and sell things in the market, and so on. They were an essential part of the functioning and survival of their households.

Third, as /u/Jepekula mentions, even the elite occasionally revealed some very forward-thinking ideas. Plato saw no reason why women should not be raised and trained in the same way as men, to fulfill the same roles in his ideal society. Aristophanes wrote several comedies in which women take the intiative to force their political views on men - making peace in the Lysistrata, and founding a sort of economic and sexual communism in the Ekklesiazousai ("assemblywomen"). Several major tragedies (Sophokles' Antigone and Euripides' Troiades ("Trojan women") in particular) are all about how women suffer when men fight and make bad decisions. Meanwhile, women actually did have some more rights in Sparta, where they could inherit and own property - perhaps a necessary arrangement given that the men were rarely at home, being forced by law to spend their days with their messmates.

In the case of Athenian, citizen-class, women took care of the household, kept the slaves in check, etc, while the man was away, in politics or doing business. I would suspect that women might have been deemed inferior to men, but Plato or Socrates, can't remember which, argued that men and women were equal in intellect. Women did not in any case have formal and direct political power, but there's the play "Assemblywomen", which you might find interesting as a scenario the author, whose name I can't remember, thought could or would happen if women gained political power.