I would argue that it is currently a big part of the current economy if you know where to look. Lots of labor works via the principles of a gift economy.
E.g., you help your friend move to a new house, they help you redo your deck, you babysit your brother's kid, they cook you dinner, etc.
The problem with bartering is that it doesn't handle 3+ way trades (i.e., person A needs something from person B who needs something from person C, who needs something from person A), and it doesn't usually handle asynchronous trades.
In gift-based systems, people can literally retire based off the goodwill that they've cultivated. There are many old people who serve their families/communities for years who then get taken care of when they need it.