The response to the above question will usually be a barter economy. But I want to show you that this is a tougher question than it seems. The answer depends on whether you're starting from a money view of the world or a moneyness view. (See Why Moneyness? in which I explain these ideas).
1. The money view, which is the standard view, begins by classifying all things in an economy into either money (M) or non-money. Any economy that has M in it is a monetary economy. All exchange in a monetary economy is achieved by trading non-money into M and back into a different non-money. When there is no M, then a non-monetary economy is said to exist. In a non-monetary economy, exchange occurs by trading non-money for non-money, our word for this being barter. So a non-monetary economy is a barter economy.
2. Things are different from a moneyness perspective. An isolated household living in a cave values their inventory of goods solely for its use-value—how each good satisfies the household's needs. The household's goods become liquid the moment that it realizes that other people are willing to exchange for them. Their inventory is now worth more to them than before because it provides them with a greater range of options. As once-isolated households trade with each other, goods will be graded according to their relative liquidities. Depending on its ranking, each good earns a smaller or larger premium over use-value. ie. a liquidity premium. Liquidity/moneyness don't exist in a world without trade. We call such a state autarky.
So depending on one's method of classifying the world, a non-monetary economy can be either a barter economy or an autarkic economy.
This conclusion may seem somewhat odd at first. Take what we would typically consider to be a barter economy, say a world in which people barter deer for beaver. This setup is actually monetary in nature, insofar as both deer and beaver earn a monetary premium for their potential to be bartered. In other words, its possible to start monetary analysis way before we ever exit from so-called barter.