Part of this may have had something to do with the expectation that Spartan men should spend much of their time off at war. Spartan women, meanwhile, were educated with the expectation that they should not only be strong mothers of strong children, but also that they should be able to be in charge of a household and presumably the land attached to it. Given that, it seems likely that most of the land belonging to men also ended up under the practical control of women for certain periods of time.
The main way Spartan women acquired property seems to have been through inheritance and through their dowries. The problem of the dowry in Sparta is a complicated one because the sources can’t seem to agree on what that meant and all of them were written by non-Spartans. Some people argue that the dowry was a gift to the woman herself on her marriage. Others would say that it couldn’t be called a dowry at all, as it was an inheritance gift to the woman and never passed into her husband’s control. No existing source tells us how this was thought of in Sparta itself.
Some women from Sparta are known to us now because of what they owned, sometimes in addition to what they did. Cynisca and Euryleonis were Olympic victresses, not because they themselves appeared at the games, but because they owned and trained horses that won chariot races. Arachidamia and her daughter-in-law Aegesistrata were the wealthiest two people in Sparta in the 3rd century BCE, giving them significant political power that women in most other parts of Greece would not have been able to hold.
*Property in Athens legally belonged to the head of the oikos, no matter who had brought it into the household. Though a woman may have had practical ownership over certain personal items, they legally belonged to her kyrios (guardian). Pretty much every example of significant property listed in a woman’s name at Athens was only held as security against the possibility that her husband might have to return her dowry.
Xenophon, Constitution of the Lacedaemonians 1.2-10. 4th cent. B.C. - Diotima
Plutarch, Life of Lycurgus 14-16, exc., 2nd cent. A.D. - Diotima
Pomeroy, Sarah. Goddesses, Whores, Wives, and Slaves: Women in Classical Antiquity. New York: Pantheon Books, 1995.
Schaps, David M. Economic Rights of Women in Ancient Greece. Edinburgh: Edinburgh University Press, 1979.
Land Ownership at Sparta - Perseus
Women in Sparta - Women in the Ancient World
The Women of Sparta: Athletic, Educated, and Outspoken Radicals - Ancient History Encylopedia
Women in Ancient Sparta - Wikipedia