In order for this to work without becoming a different form of money, it sort of has to happen on a small scale.
Say you mow my lawn, with the understanding that I'll help you bring in your wheat. I might not be happy if you trade that favor to your neighbor, and then he comes and asks me to clean his pig pen.
Likewise, I don't know if I'd accept a traded promise of a favor if I don't know the person who's supposed to be doing the favor. What if they aren't that skilled, or won't keep their word?
A favor based system would probably work fine for most stuff on a one on one basis (you help me and I'll help you, if you don't keep your word I'll remember), so the big question is how to take care of the big stuff, like keeping up the roads and stuff that are usually take care of with taxes?
The short answer is that communities would have to take care of it together with each person pulling their weight. Work parties and community events. Every one goes out and fills in the ruts, or helps the people filling in the ruts, driving in gravel, carrying water, cooking food, etc, and afterward you have a feast with dancing and singing.
A favor based economy would really be a reputation based economy, with each persons reputation buying them what they need, and if they spend all their reputation without doing things to replenish it, they would be broke.
Edit: More thoughts on reputation/favor based economies...
Say you move to a new area where no one knows you, you'd basically be broke. There might be some people that would do you a favor on spec ("You look trustworthy, I'll do this for you"), but for the most part you'd have to build up your reputation capital from scratch. It wouldn't have to take long. Things like community work where you can show your work ethic, doing small favors just to help people out, you can get a reputation as a good guy and a good worker real fast.
The other way is if you had references ("Sir, I don't know you, but I know Bob, and he says you're a good guy, so that's enough for me"). This would work very similar to lines of credit. Bob is known to have good character, and he vouches for you, so people of the new community would be more willing to help you out. However, if you don't end up pulling your weight then those lines of credit would dry up, and Bob could lose credibility in the process.
A favor economy works best if you don't keep score. I could do you several small favors without asking for one in return, just because I don't have any needs. This could start to make you nervous, wondering if I'm saving up for something big, like asking for help hiding a body.
The best case is when I don't keep track of all the times I helped you, and you help me when I need it just because we're neighbors and I've helped you in the past.
The exceptions are bartered favors (help me roof my house and I'll help harvest your field).
The one thing I'm not sure how to resolve is land. In the Long Earth land isn't an issue because you can step sideways and have an earth all to yourself. With out a long earth you could abolish almost all scarcity except for land... I don't know how property purchasing would be figured out, how to determine who gets beach front and who gets landfill facing... etc.
Edit 2: Why is this not currency?
With a currency everything is given a number. This unit of work is worth this amount of money, and this other unit of work is worth a different amount of money. This is not a criticism of money. Money is a great system. It lets me, a programmer, hire a plumber to fix my pipes, when he doesn't need anything from me. Otherwise I'd have to find out what the plumber needed, and then find someone I could to a favor for that that could in turn do what the plumber needed. That starts to get complicated.
But OP didn't ask about how to design a new currency, he wanted to know if a economic system could be based on favors only, without any currency.
The answer is it could, assuming that people are selfless enough.
It would rely on people wanting to help out others to the point of inconveniencing themselves, with the hope that the other person would do the same some day.
It would be easier if it was mostly a post scarcity world like in the question, since then people wouldn't be giving up stuff other than their time.
If I only have 5 chickens and you want one that might be a hard thing for me to give as just a favor. (maybe as a bartered favor?)
But if I have a lot of chickens then it stops being a big deal.
The other thing that makes reputation different than money is that I can be a pretty horrible person and have lots of money, and buy all the things.
Being reputation and favor based would give me an incentive not to be a jerk.
Likewise, I could have a pretty good reputation with almost everybody in town, but a bad one with my neighbor because my dog keeps getting into his yard. It's not something you can really put a number on, it all has to do with personal relationships and feelings.