On paper, this sounds a bit like delayed barter, but it bears some significant differences. For one thing, it’s much more efficient than Smith’s idea of a barter system, since it doesn’t depend on each person simultaneously having what the other wants. It’s also not tit for tat: No one ever assigns a specific value to the meat or cake or house-building labor, meaning debts can’t be transferred.
Anthropologists have argued, in contrast, "that when something resembling barter does occur in stateless societies it is almost always between strangers, people who would otherwise be enemies." Barter occurred between strangers, not fellow villagers, and hence cannot be used to naturalisticly explain the origin of money without the state. Since most people engaged in trade knew each other, exchange was fostered through the extension of credit. Marcel Mauss, author of 'The Gift', argued that the first economic contracts were to not act in one's economic self-interest, and that before money, exchange was fostered through the processes of reciprocity and redistribution, not barter. Everyday exchange relations in such societies are characterized by generalized reciprocity, or a non-calculative familial "communism" where each takes according to their needs, and gives as they have.
Another advantage of bartering is that you do not have to part with material items. Instead, you can offer a service in exchange for an item. For instance, if your friend has a skateboard that you want and their bicycle needs work, if you are good at fixing things, you can offer to fix their bike in exchange for the skateboard. With bartering two parties can get something they want or need from each other without having to spend any money.
It’s hard to answer that without actually seeing a modern gift economy in action. Luckily, modern gift economies actually do exist. On a small scale, they exist among friends, who might lend each other a vacuum or a cup of flour. There’s even an example of a gift economy on a much larger scale, albeit one that’s not always in operation: The Rainbow Gathering, an annual festival in which about 10,000 people gather for a month in the woods (it rotates among various national forests around the country each year) and agree not to bring any money. Groups of attendees set up “kitchens,” in which they prepare and serve food for thousands of people every day, all for free. Classical economists might guess that people would take advantage of such a system, but, sure enough, everyone is fed, and the people who don’t cook play music, set up trails, teach classes, gather firewood, and perform in plays, among other things.