Not really. You can't get away from "accepted market value". It falls out of a small handful of simple axioms. See something like section 2 of http://www.daviddfriedman.com/Academic/Price_Theory/PThy_ToC... , which shows (in several steps) how money falls right out of the way people have different valuations for different things.
Barter economies are degenerate cases of market economies, where the transactions costs are higher due to the nonliquidity of the currencies actually being used. This is not a story about a beautiful proof of how money isn't necessary, it's proof that the money-based economy is rich enough to support not-100%-efficient sub-economies as effectively parasites on the main economy. Still true, but a lot less faux-beautiful. Money is still there, just swept under the rug so everybody can feel warm fuzzies about not using money.
Note that this is a separate issue from asceticism; market economies aren't incompatible with owning stuff, rather than stuff owning you, and bartering doesn't enforce asceticism.
The way I see it is there are definitely still markets - it's just that each relationship between person A and person B is a unique market with unique market values for goods and services. "MarketSpace" A->B would actually be different than B->A - so a theoretic ideal situation would be where "arbitrage" ends up benefiting both parties because of their relative markets.
Example:
I can re-build volkswagen/porsche engines. While the market value for this is roughly 1.5k usd in labor I would gladly do it in exchange for you staining my arcade cabinet - a service with a roughly 300-500 usd value.. I would feel like I was getting a killer deal and so would you.
Is that "degenerate" because it is not optimizing exploitation or do you mean that in another sense of the word?
It's mathematically degenerate for the reason you mention; between every pair of people, you have an independent "market". This greatly limits the ability of the markets in question to do anything remotely economically efficient. Follow the link I posted and read for a while, and it'll take you through the process of deriving the concept of money from a set of those markets. (Well, sort of; it comes at it from a different direction but it's equivalent.) The real world is more complicated.
Incidentally, the model in that book will be mathematically degenerate in the other direction, at least for most of the book, because it goes to the opposite extreme of completely ignoring transactions costs for a while. Reality is, as usual, somewhere in between, but it is much closer to the market-with-money model than the everybody-is-bartering model.
So, I'm using it in the math sense, not in a moralistic sense.
If the phrase "market efficiency" is striking fear in your heart, remember that what that means in reality is that the barterers are closing themselves off to better deals and reducing their wealth vs. what they could be doing.
So, if that's true, why are they bartering? In the real world, money does involve a certain floor of transaction costs, imposed by the government for tax reasons. (I'm not saying I disagree with this, just that there are costs.) Things near the boundary of those costs can be worth just bartering, or selling "under the table". But it's effectively impossible to "barter" your way to a car or a house. (Yes, I know it's been done but it's generally a gimmick.) It can also be possible to barter between two people who simply have no money, in which case the reduced efficiency is a tradeoff you have no choice but to make. (One of the worst things about being poor is how it can lock you away from the very higher-efficiency economic transactions that you need to escape from poverty.) But like I said, it's not the awesome blueprint to a new economy (which I realize wasn't explicitly said in the article), it's a special case that has little general application.
Barter economies are degenerate cases of market economies, where the transactions costs are higher due to the nonliquidity of the currencies actually being used. This is not a story about a beautiful proof of how money isn't necessary, it's proof that the money-based economy is rich enough to support not-100%-efficient sub-economies as effectively parasites on the main economy. Still true, but a lot less faux-beautiful. Money is still there, just swept under the rug so everybody can feel warm fuzzies about not using money.
Note that this is a separate issue from asceticism; market economies aren't incompatible with owning stuff, rather than stuff owning you, and bartering doesn't enforce asceticism.