The best evidence for how ancient Greek textiles were produced would be the textiles themselves. Sadly such textiles are very rare. But we can also learn about ancient Greek extiles by examining the tools involved in their production, such as spindles, carders, looms, shuttles, weaving swords, bobbins, dye vats and beaters. While no ancient Greek looms have yet been discovered, some components of looms - such as loom weights - have been found.
Both looms and textiles were made of organic materials such as wood, wool and other animal or plant-based fibres. Only under very special circumstances do these materials survive over time. But loom weights made of stone, metal or baked clay can and do survive in the archeological record, as do spindle whorls and bobbins made of the same materials. While clay can dissolve, crumble or break apart, if the clay is exposed to fire (either deliberately or accidentally) it can harden enough to last thousands of years.
Fortunately, in the case of ancient Greece, there is also other evidence about textiles and textile technology. There are references to both textiles and textile production throughout Greek literature; and there is pictorial evidence, found on vases, wall paintings, and in one case on a marble stele (Athens, NM1914).