Children of the Nile is a video game which is set in a real-life environment, where barter is used. That is not an RPG, though.
Keeping a tab at the shops of various sellers is the first refinement of the barter system. In that system, each shop accepts any goods of worth that I happen to have. In return, I get whatever goods that shop keeps in stock. Then the shopkeeper has to figure out how to convert metal scraps into flour, for example. Possibly he can trade the dented armor to the blacksmith for knives. Lots of people would probably accept knives as trade goods.
The next step is paying for goods at the above shop with credit chits from the blacksmith, obtained by dropping off the metal bits at that place. Now we have currency, issued by every shop, or at least by major shops. Then it becomes currency issued by the local ruler, to consolidate the value of it and incidentally make taxes easier to assess and collect.
Now we have modern times, and our trade is carried out using a mix of issued currency, credit chits issued by banks (checking), and tabs kept also at banks, and sometimes at shops (credit cards). We've progressed almost back to before currency again by not needing to carry individual items to represent our buying power, but the concept of relative values is still there. It always was. One mammoth steak was always worth two frozen yucca roots, or whatever.
Which is to say that we always have some kind of numerical valuation in mind even if we don't specifically assign numbers to everything before we trade. The kid who trades his peanut butter sandwich for a pickle doesn't assign explicit numbers to the two items, he just knows that he wants that pickle more than he wants the sandwich. I think that would be a weird trade, but then I'm not that fond of pickles as compared to peanut butter. *laugh*
So, it should be possible to write an RPG with nothing but the purest barter visible to the player, even though the game must make numerical valuations in order to present the proper choices. Bandits would be stealing anything of value, not just gear. They would not attack people who are better armed than themselves, for the given reason that they would die, but they could prey on weaker victims. Knives, leather boots, personal jewelry, and anything else that could be bartered would be fair game for them. There would not be the concentrated value of leather pouches of coins, so probably cutpurses would not flourish, but a dead body can always be stripped of everything and provide some kind of loot.
Edit: I seem to recall that some communities in the United States have attempted to re-establish barter in an attempt to circumvent paying taxes of all types. I'm not sure where, but for some reason I think it was in New England. People were getting "paid" in barter points and trading those in to local shops, who in turn "paid" their bills back to the companies who had hired their customers as workers. I can't remember how well it worked, or if the goverment(s) figured out a way to tax them regardless, but I'm not betting against the government's persistance on that issue.