People who say "we need a world without money" are actually saying "we need a world without trading and barter," for the two are the same.
Money is simply the good for which people most commonly barter. Remove any form of money from society and you'll quickly observe people finding other things to trade with.
If trade and exchange are occurring, then goods are being traded between people. Whichever good is most commonly used in trade is given the name of money. Not only is it a silly idea to "remove money" from society, but it is in fact as impossible as removing speech.
People here underestimate the intelligence of people who advocate a world without money. I'm one of those people btw, although I see implementing that kind of system as a very long term plan. Bitcoin is a step forwards and it's very practical, which is why I like it. It's something that's happening now, not 50 years in the future.
You are correct to say that people who say they want a world without money mean they want a world without trade and barter. What you don't understand, and I'm not surprised, is that there is technically nothing stopping us from doing just that. Barter could obviously exist for unique items but technically we can produce everything people need in enough abundance for the whole world. The resources are there and the technology level is there.
It would require some changes to way we produce and use things, production would need to be more cyclical, creating products to last and focusing on recycling more. And more than that, private property would have to be rethought. The focus would need to be on access which means that we would make sure people can get conveniently from point A to B, for example by an advanced carpool which could be an automated taxi system and of course with automated mass transit that's already in use, such as subways.
No one would own these transit methods, they would simply be there for everyone to use. This is gargantually more efficient than the disastrous system we use right now where most cars have a single person in them at a time. And even worse, the cars are parked most of the time simply taking huge amounts of space. In an efficient system we would only have a very small percentage of cars parked, for a buffer. This same principle can be used for a number of things, everything that people don't constantly need can be "pooled". With these efficiency changes we can provide basic needs to all 7+ billion people if we want to.
It isn't straightforward to remove money even if this was chosen as the proper direction, money has radically corrupted our incentive systems which is one of the main obstacles for a different kind of system. But I think that it won't take very long anymore for automation to get cheap enough that companies don't have any cheap country to go to for cheap labor, they will simply build robots. That is the breaking point for our current system, because there won't be enough work for people anymore.
Then we are forced to give people the purchasing power they need, which changes the game entirely. This development could eventually lead to a system with no money because the incentives to do stuff would be based on creativity and intrinsic motivations. Extrinsic incentives and rewards based on them are on their way out, once the breaking point is reached the type of system we have now is simply over.