This situation works because you have something that substitutes for cash in your community... trust.
Cash is a way to create trust between two parties that don't know each other.
Without cash, if I walked into Best Buy for a TV, I'd offer to trade goods or services for it. Since Best Buy doesn't know me from Adam, they have no way to know whether I'll make good on my promise (they can't trust me), and so giving me the TV is a risk. However, Best Buy probably trusts the US Government to continue accepting dollars for tax payments, so dollars are valuable and trustworthy. If I give Best Buy some dollars, they are immediately useful in paying taxes.
Likewise if I work for Best Buy, and there was no money, Best Buy would 'pay' me by providing everything I ask for up to a certain standard of living. I'd have to trust Best Buy not to abuse this situation. It's far easier to be paid in dollars, because I can trust that businesses I buy from will value dollars... because they can trust that the government will value dollars (for tax payments).
The only way to survive without trust (either the real thing or via cash, which is really trust in a third party), is for everyone to be totally self-sufficient. Specialization is a huge economic boost, so giving it up means a lower standard of living for everyone. However, if everyone were truly self-sufficient, there would be no need to trade goods/services, and hence no need for trust or cash.