1. If one set of goods or services are considered to have a different value from some other set of goods or services, and if it's possible for people to trade goods or services, congratulations -- you have an economy.
It may be grotesquely inefficient (communism); it may be brutally efficient (fascism); it may not stimulate entrepreneurialism (socialism); it may not be perfectly fair (capitalism); but those are all degrees of effectiveness. If people are trading differently-valued things, it's an economy.
2. Barter is hideously inefficient as a means of exchanging wealth-tokens. Any civilization worthy of the name will immediately begin minting coinage in order to use currency in place of goats and chickens.
Of course, if you're a Yap Islander, your coins might not be terribly portable. But it's still easier than swapping goats and chickens.
3. A highly advanced civilization, in which the most valuable asset is imagination (i.e., intellectual property), is incapable of being based on barter -- there simply are not enough goats and chickens to go around. Even more profoundly, the moment your monetary system migrates away from minted coinage to ones and zeros in a computer database, the amount of "money" that an individual or a nation/planet/Federation can control increases exponentially.
In fact, I would guess that a Galaxy-class starship could never be built in a barter economy. Such things (to say nothing of planetary-scale Marshall Plans or similar massive public works) would likely be so expensive that only a financial system based on the abstraction of digital money could afford them.
I've said before that it wouldn't bother me if a Star Trek MMORPG didn't use money, and that's still true. For one thing, whether we like it or not, the whole "we don't use money" thing is canonical; for another, I think not having currency would help Star Trek Online: The Game cut down on gold farmers.
But if we're looking at Star Trek: The Future Human Society as though it were real, then "no money-based economy" is just not viable. For that to happen, either all goods and services would have to be considered to have equal value, or the desire or ability to trade goods and services would have to be eliminated completely.
Either of those things might happen... but we'd no longer be human.
And I don't think even Star Trek goes that far.