Certainly not like the 300.
The Spartans did not dress differently from the other Greeks and their costume, despite changes over time and social differences (which existed despite propaganda), was simple. The most common was the Doric chiton, a square-shaped tunic made of wool or linen (depending on the season), which was placed around the body and fastened with buttons or brooches. Around the year 400 before our era, the exomis became fashionable in which a sleeve, the right one, was dropped to leave the arm free.
On top of the tunic they wore a red cape, which was one of their favorites; because, among other things, it hid the blood. There were, not only in Sparta, two models of cloak; the himation and the chlamys. The former wrapped the body, with or without a tunic underneath, and the chlamys was more like a cloak. The Spartans used a characteristic type of himation, the tribo that was shorter than those of the rest of Greece.
The footwear, when they were not barefoot, which was the most common, was a sandal or a leather boot, the latter open to facilitate movement, it was called haplai, with an opening in the heel.
Another element typical of the Spartans was the bacterion, a staff with a T-shaped handle, although they used it as a support on the trails and, as in the case of Euribiades, to hit an inopportune Themistocles, the bacterion was an emblem of Spartan emissaries.
In war they wore armor, even when it was hot, identical to the traditional panoply, namely; breastplate ( thốrax) of bronze that sometimes imitated the shape of the muscles, the protective cnemids of the tibias, the bronze helmet with protections for the cheeks and the aspis or shield of wood covered in bronze. The whole armor used to weigh about 20 kg and the shield alone, about ten. In the Spartan tradition, throwing the shield (something essential to flee) was the greatest shame, hence the phrase: “with him or on him”. In classical times the armor was something that each hoplite had to pay for himself and there was no uniformity, on the contrary, the Spartan hoplites had panoplies paid for by the state and made in series. The hoplite shield is said to have been marked with the letter Λ, for Lacedaemon, but this seems to have been in times after the Persian Wars.
A distinctive feature of the Spartan, rather of the Spartans, full-fledged citizens, was long hair that they combed for battle, they also wore a beard.
During the trips the use of the hat was common, the one with wide wings, or petaso was little used in Sparta, the pilos, a conical hat that usually appears in the images of the Dioscuri was more common.
The women used a chiton, like all the Greeks, but less heavy since they used to exercise with it. Before using the chiton, like all Greeks, they wore a peplo, which was rectangular, sewn and folded.
As for underwear, women wore a linen band over their breasts called a strofion, both sexes wore a perizoma, a knotted cloth like the one that appears in the crucifixion scenes, as underpants.
I add some illustrative images