Why would an economy without money not work? Just because we and our economic elite are so entrenched in money and free market theories that border on religion that does not automatically mean that other ways of organizing a civilization do not work as well. If you pulled a Roman citizen off the streets of Rome and told him/her that in 2000 year or so people will buy silk (a very expensive luxury back then) with something resembling papyrus money rather than solid gold aurei he/she would have either laughed at you or if they were a kind hearted person offered to escort you to the temple of Apollo so that you might have your lunacy treated by a skilled healer.
Your example however fails to demonstrate that an economy without money would work. While the Roman would have laughed at you or pitied you - only the medium of exchange has changed. The underlying organization of the civilization and economy remains the more or less same, and still relies on money.