I think the best way to think of this is not as something that started in the future; it really started in the industrial revolution and just kept going.
In the early 1800s there was a group in England called the Luddites, who went around smashing machinery they thought was taking their jobs. The term is even synonymous with someone who doesn't use/like technology. And the thing is, there's a lot of evidence that those machines really were taking their jobs.
But far from being the end of the world, the labor market slowly adjusted. Some people lost their jobs, but others got new jobs that hadn't existed before. On top of that, everyone got more and cheaper goods.
The world you're describing would have been the result of a very long, multi-century (or millennia if you're far enough in the future) process. There would have been lots of periods of adjustment, likely often accompanied by upheaval such as the Luddites, or the decline of the middle class we're glimpsing now.
Labor as an Exit from Poverty
One thing your world might suffer from is a lack of a way out of poverty. In the US, low-skill trades like the auto industry were a path to the middle class for millions. Later those jobs moved to places where people work for much lower wages. There's actually some evidence in development economics that China is currently "clogging" this path for everyone else; if you're a small poor country somewhere interested in expanding manufacturing for your people, how are you going to compete with China, and particularly their infrastructure?
So maybe in your world no countries that didn't already escape from poverty can find a way out. They're just poor countries that stay afloat on volatile commodities markets and international aid - only now the thing blocking their way out isn't China, it's that the entire world has cheap robot labor.
If robots are so good at everything that they've entirely replaced low-skilled labor, it's likely that manufactured stuff is cheap. You say people still need to work to get by, but how cheap would it be to send a bunch of construction robots over to whip up a new tenement when needed? They don't need breaks or lunch hours, don't go on strike, presumably make near-zero mistakes, and don't particularly worry about their own safety.
Are massive farming operations completely automated by these same tireless robots? What does that do to food prices? And so on.
It seems very likely to me that a subsistence level of living would be widely available. Large swaths of people would probably be miserable due to a lack prospects for upward mobility, but they almost certainly don't lack for basic necessities.
Shifting Labor Market
This has been mentioned in a few other places, but it's possible that the fields that machines aren't able to do well, like music or art, would just... expand. If there's tons of wealthy people who have easy access to the things that robots can do, what do they do with their disposable income?
Hire yourself an artist, or spend way more time going to things artists do. Get a nanny, or heck, get two. Maybe machines can handle fairly boring food for everyone, like going to eat at Ruby Tuesdays or something, but what people really want is their own human chef to show off to their friends!
Or maybe even new jobs we can't imagine? Could Luddites have envisioned computer programming jobs? Maybe it becomes en vogue to have your own personal self-esteem person, who follows you around and tells you how great you are at every turn.
Think of productivity as the number of people working it takes to support everyone. Our standards of living can increase across the board as long as our productivity increases proportionally. That is, if some new innovation makes everyone twice as productive, we can basically double living standards for everyone.
Likewise, we can reduce the number of people working as long as the people who are working are proportionally more productive. That is, if some new innovation makes everyone twice as productive, we can maintain living standards by having half as many people work.
Then imagine how your society falls on this scale. Essentially robots have made productivity go off the charts. How that productivity gets used is what will dictate your answer.