Last year, RAT member Ivor (Crispianus) alerted me to the book Leatherwork from Elephantine (Aswan, Egypt): Analysis and Catalogue of the Ancient Egyptian & Persian Leather Finds. Among other useful information, you can see on page 102 (104 of the PDF) an Achaemenid-period bag in the form of a cylinder with a circular bottom and drawstring top. While it may or may not be Persian, it at least gives solid evidence for a kind of purse or pocket that was carried during our time and within the borders of the empire.
A reenactor needs a small bag that can be kept on one's person for holding phones, money and other possessions that are best not left unattended around camp. This bag is just about appropriate for that purpose. It should be easy enough to make a replica should you have the time to do so, but similar bags already exist for the reenactment market.
One that caught my eye earlier this year is marketed by Crazy Crow Trading Post: the large ball bag. (British readers may pause to finish laughing before reading on.) Its basic structure and even its size are similar to the Elephantine find: 6-1/2 by 4 inches (165 by 102mm) versus the Egyptian bag's 140 by 95mm. The Egyptian bag is sewn with linen. Its leather wasn't tested for tanning technique, but it has the color of and somewhat looks like smoked wet-scrape braintan. The Crazy Crow bag is most likely made of a chrome-tanned cowhide split, dyed to resemble smoked braintan. I have no idea what the stitching is made of.
There are some minor construction differences: The Egyptian bag has two side seams where the Crazy Crow bag has only one, with thin folded rands on one side seam and around the bottom seam which the Crazy Crow one lacks entirely. The Egyptian one also has a pattern of punched holes near the top, which could of course be added to a commercial bag after purchase if you have a small awl.
In person, the Crazy Crow bag is made of medium-weight suede, rich golden (a color that can be achieved by smoking undyed leather), with a much shaggier surface than would be produced by wet scraping. The long drawstring is made of the same leather, which is thankfully not very stretchy. Note that the catalogue description implies that these bags are made of various leathers - I expect they use whatever they have on hand that works - so the exact weight, color and texture will probably vary somewhat from one batch to the next while still fitting the general description of moccasin-weight cow suede.
As for practical use, I selected and borrowed some items that you might want to keep close to you at an event: a 5-3/4 by 2-7/8 inch (147 by 73mm) smart phone, wallet, coin purse holding a few emergency medicines, roll of 10 bills and key ring with various mechanical and electronic keys.
The answer: just about. The drawstring almost entirely hides the top of the phone. You would probably also be able to cram a passport in there.
Conclusion: This isn't a top-of-the-line item, but it will certainly do if you're pressed for time to construct one that's a little more authentic.