People in ancient Greek cities like Sparta and Athens got water from, well, wells. When I was excavating the ancient Greek city of Sicyon we also did some work at the well of the agora, just a few meters from the buildings that we were digging. And remember the “this is Sparta” scene in 300? It was based on a real incident and it involved a well.
And it wouldn’t be just a well in the center of the city. There would be many. My father’s home at a village on the mountain had an well as many old houses did. Although it’s in a small village there used to be 2 public fountains with drinkable water that still work, the building around them built in the 18th century. Even in a smaller village on our way there we used to stop and fill bottles with water from a fountain by the road. One may find such fountains in many villages, for example here’s the fountain at the central square of another village of Pelion, near my house at Volos.
There were plenty of springs and ancient Greek cities would be built in places with underground water, making wells and fountains at the springs. Also cisterns were a thing. This has been a common thing to watch again at the villages of my parents, especially in farmhouses by the slopes. A small stream or a spring or just rainwater would fill a cistern and that was useful in premodern Greek houses. And as for ancient Greek cities Pausanias mentions such things in his Description of Greece.
In Athens he mentions springs and cistern all over the city and a fountain built by Peisistratus. In Corinth he writes that that there are many wells and that the city had plenty of water even before emperor Hadrian brought water from a nearby lake. Even small ancient Greek cities such as Hermione is mentioned to have several wells old and new. The small Phocian city of Lilea had baths and plenty of water as it had springs from which the river Cephisus began. The rule was for Greek cities was to have lots of wells as Pausanias mentions that the Phocian city of Stiris being in a high and rocky place had few wells with water good only for washing and for animals, the drinking water provided by a spring 700–800 meters further down.
Fountains seem to be found in many Greek cities besides Athens. Pausanias mentions fountains in Megara, Corinth, Mycenae, Epidaurus, Troezen, Sparta, Therapne (also a fountain on the way to this Lacedaemonian town), the ruined city Cyphanta, and Messene, at its agora,
(which reminds me I should visit it)
The people of Corone get drinking water from a fountain in a plane tree. Also fountains are mentioned in the Arcadian city of Aliphera, another outside Tegea, where he mentions another fountain too. In Thebes a fountain is mentioned at the site of the battle against Alexander and some other fountains too are there. There’s a fountain at Orchomenus mentioned too. It was pretty much a rule for a city to have such things that when Pausanias speaks of Panopeus, a city in Phocis, he is sarcastic about calling it a city since it lacks the basics such as gymnasium, theater, agora and no water descending to a fountain. Apparently it had water but they weren’t “civilised” enough to have a fountain. Another town in Phocis, Hyampolis, is mentioned to have only one well and this is mentioned as an oddity as it was their only supply of water besides winter rains. Here emperor Hadrain only built a stoa, no aqueduct.
This brings us to the false premise of the question.Greeks did get water through aqueducts too. No fancy bridges etc over huge distances but still impressive aqueducts occasionaly, such as the one in ancient Samos.
This was built for the ancient Greek city on the island of Samos way before the famous Roman aqueducts. Pausanias also mentions the city of Kirra being provided water through a channel they had made which brought water for the nearby river Pleistus. So, although ancient Greek cites as a rule would have water from wells, springs, fountains, etc they would occasionally make a channel or an aqueduct to bring the necessary water.