For the most part the sentiment is accurate, but not for the ideals of feminism that we hold today.

The women in Sparta during the classical period did enjoy many freedoms compared to the rest of the Greek World. Women were able to purchase and sell property, enjoyed access to education, and were able to exercise/go about Sparta publicly with no male supervision. The reasons behind this primarily lie behind Spartan culture as a whole. From around the age of 7 Spartan males were required to enter the agoge, living in the Spartan barracks for most of their adult lives. Because of the men's constant focus on warfare and the continued subjugation of the helots, it fell to the women to manage the home estates, roles that would have been controlled by men in other Greek cities. Women in Sparta also did not spend most of their time weaving like other Greek women as this was primarily done by their Helot slaves. The Spartan woman's focus on exercise and education originated in the Spartan's early view of "eugenics". Spartans believed that the more physically fit and skilled the mother was, the likelier the children born to her would be healthy and grow to be strong warriors. Spartan girls' education primarily consisted of "mousike", an education focused upon subjects such as dancing, singing, and gymnastics. Much of the musical aspects of a Spartan girl's education came from Alcman, a possible Greek slave or traveller, who composed many songs and led the Spartan girls' choir. However, all of these privileges were only accessible if you were a Spartan woman, a group that was a very small minority in Sparta. Other groups such as the Helots were treated as a virtual slave race, and the peroikoi who were a group of people living in the hills of Laconia but did not hold citizenship in Sparta.

Over at Athens, the polis may have had a democracy, but this only was a privilege for male citizens. The Athenians were very conservative in regards to how they treated their women. Women were not allowed to be in public without male supervision and were not allowed to attend plays at the amphitheatre. Athenian men often would not speak of their wives in public unless it was to a relative or close friend so that they would not bring dishonor on their wife's name. Men in Athens were allowed to sleep with unmarried women but not another man's wife. The women on the other hand were only allowed to sleep with their husband. An Athenian women would typically manage the domestic affairs of the house but could not own or sell property. The dowry of an Athenian women would often pass from her father to her husband in her name, and the items a woman could own were often only transportable goods. Athenian women had no formal education but usually were taught how to weave and manage a household by their mothers.

I apologize since I am on mobile so I was not able to be detailed on quotes further elaborating the subject from people like Xenophon and Aristotle

Sources: Spartan Women by Sarah Pomeroy The Spartans by Paul Cartledge