Money is such an obvious and natural concept, that it effectively can't not be invented.
There are roughly three kinds of money:
1. Commodity money is just any suitably rare commodity. It must be sufficiently hard to get that it maintains value, but still common enough that everybody may possess some for the purpose of exchange.
Precious metals or gem stones are most suitable, but (as mentioned by the other answers) pepper seeds, pelts, bird feathers, sea shells, carved stones and, in absence of anything better, even cigarettes were used as money. And there is really no distinction that would make gold more of a money than cigarettes!
It is unimaginable that a global society would not get at least to this stage. Basically every human society with non-trivial organisation in history invented commodity money, often independently.
1. Representative (commodity-backed) money bring the idea a step further. Instead of carrying the (potentially heavy), there is a storage that gives out notes that certain amount of the commodity stored there belongs to the holder—a bank.
It is a slightly less obvious idea, but still easy to establish, because the first bank can be easily established by anybody with the right idea. As long as the bank notes can be exchanged for the real commodity, it does not require any other backing, agreement or government enforcement.
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Fiat money does require general agreement or government provision. However, once most trade switches to representative currency and nobody exchanges it for the commodity it represented any more, dissociating it from the commodity becomes the obvious step.
First the banks—and more importantly the governments—will get a growing itch to give out more money than they have of the backing commodity, relying on the fact that almost nobody converts it any more. And when the disproportion grows too big, they'll cancel that promise—and everybody gets stuck with the now completely free floating fiat currency.
This is relatively recent evolution. US Dollar was commodity-backed until 1971.
Now free floating fiat currencies have their advantages. Their amount adjusts to the economic performance, which smooths out price fluctuations and somewhat compensates for regional differences. The main disadvantage is that they permit hyperinflation.
Economy that didn't evolve fiat money will be somewhat less efficient. The amount of commodity(-backed) currency is governed by production of that commodity and not needs of the economy, which leads to situations where there is too much or too little money in some region and that in turn leads to large fluctuations in prices. That makes planning harder and forces long-distance traders to work at higher margins, which is what reduces the efficiency.
So without modern fiat currency, international trade would exist, and would quickly settle on one or a few commodities that would become currencies. Expect it to be somewhat less efficient, and expect it to make the good trading companies and centres disproportionately richer (similar to how it was in middle ages) as they would be bearing the rate risks and would get paid for it. 130