If there were no currencies, then how would ownership work? What gives you the right to own something if you haven't done anything to earn it? If you haven't paid for it, then it's hard to just claim it. You might say, "But people would just say 'This is mine,' and it would be okay." It's true, people would, but that wouldn't work either. What if somebody just said "Everything is mine," and claimed everything for their own? Then you could say, "Well, we could set up a system where people can only own a certain amount." You could, but how would you compare the value of a biscuit to the value of a Porsche? Would they be worth the same amount? No, of course not. So, you'd give them a value based on the nature of their use, the availability, and other things.
So, people would then be assigned a certain amount of "Ownership Points". But, to get these points, people would have to do things, and the more they worked, and the more value their work had, the more of these points they'd earn.
Oh, hang on. You receive valued points, redeemable for things, that you have to earn through work? Sounds like a system I know...