Money is fundamental.
I'd say private ownership and trade is fundamental, inasmuch as - regardless of the society - it arises spontaneously and can only be eliminated by force, but money itself is only "fundamental" to a certain kind of society. A society with a large population, an active economy, and easy communication over long distances could not plausibly be sustained without money... but theoretically if we were willing to go back to the bronze age, money wouldn't be necessary.
Even in the bronze age they had the concept of money, I think. The romans I'm sure they did.
Only a very primitive, aborigine-like society can live without it. And in such societies, you won't have true private ownership nor trade.
It's interesting though, because most of our "way of thinking" comes from the times when our ancestors used to live like that. No wonder why so many people seem to hate money, profit etc, as well as think the economy is a zero-sum game. It was indeed from a zero-sum world that we evolved. A very interesting text on this subject: http://mises.org/daily/4700