The myth implies that this is a facet of the most skillful weaving and the purple cloth of Arachne's father implies that it's an expensive craft. Is weaving scenes and pictures (beyond simple designs and borders) known to have ever been part of Greek textile production (even just for the elite)?
If yes how were these fabrics used, as clothing or decoration?
Unfortunately there have been very few textile fragments discovered from Ancient Greek cultures, though the Hallstatt mines have revealed much with regards to how fabric would have been produced at the time in a neighboring area. What we do have are depictions in literature and on pottery.
Some descriptions of women weaving story cloth, besides Arachne, come from both the Odyssey and the Iliad, where Penelope and Helen engage in this form of weaving. The specific type of cloth would have probably been a weft-faced cloth, rather than a warp-faced one, since it is more flexible to changes in the pattern, or corrections of mistakes. (Edmunds says Helen’s story cloth is described as diplex, or double cloth, but then goes on to say double cloth can only reliably produce simple patterns like blocks or stripes.) It is, as one could guess, extremely time consuming, so the depictions of aristocratic women weaving this way indicates that is was probably a class restricted art form. (Side note: Helen also was said to use a golden spindle, and gold spindle whorls do show up in the archaeological record.)
One fantastic example of story cloth reproduced on pottery is from the Krater of Kleitias and Ergotimos, in the Museo Archeologico Nazionale in Florence. One of the Fates wears a dress with repeating bands of people on horseback and in chariots. Two more women on the krater wear similar dresses showing people on horseback and on chariots, one while riding in a chariot herself.
So, this cloth could, and was, definitely worn, but that doesn’t negate the possibility some story cloth being produced for display only.
Sources:
Elizabeth Wayland Barber. Women’s Work: The First 20,000 Years.
Ellen Harlizius-Klück and Giovanni Fanfan. “On weaving and sewing as technical terms for ancient Greek verbal arts.” https://classical-inquiries.chs.harvard.edu/on-weaving-and-sewing-as-technical-terms-for-ancient-greek-verbal-arts/
Susan T. Edmunds. “Picturing Homeric Weaving.” https://chs.harvard.edu/CHS/article/display/4365
(Apologies for any formatting issues as I am on mobile.)