Yes, it is. In a world wherein people barter, you would find them inevitably tend to barter with common goods that were marginally more universally desired. If you want the wheat I'm selling, and I want something from you, I will tend to desire a good that you have which I know I can trade to someone else. That means, I'll pay you a higher quantity of wheat in return for, say, some lumber planks as opposed to your collection of paintings. Why? Because the lumber planks are more easily tradeable to someone else.

As this process happens, a natural price-differential occurs favoring those items which are more easily traded. Eventually, a few of these items become so universally traded (whether lumber planks or rice or seashells or gold) that BOOM they are now the money of that society.

"Money" is simply the most successful barter-good in an economy. It is thus impossible to remove "money" from society without removing trade entirely.