Would the world be able to function without a monetary system (money)?
No … but today’s world would never even attempt to function without systems of money, either. Systems of money develop of their own accord, which I don’t see any entity preventing. So, unless Alien Overlords took over the Earth and dictated that money should no longer exist, I don’t see how it would happen that the world gives up all forms of monetary exchange. Perhaps, someday in the future, humankind on a large scale will somehow evolve beyond greed and vengeace and envy, and be able to achieve some form of communal society where everyone shares everything and works to the best of their ability …. but that is certainly not the world of today.