I earn a salary, but, like many American workers, I don't spend all of it -- I stash away some in the bank. This is a good thing, since I can one day retire (stop producing), and still live comfortably.
Suppose everyone succeeded in this goal. Could such a thing be possible?
Reason for no: everyone can't save money at the same time, because money is only obtained by taking it from someone else. And printing money into the economy inflates it such that it doesn't actually bring inherent value.
"If everyone can profit" could be taken by if everyone has access to a chance to gain something from an investment. This investment can be of one's own labor (e.g. harvesting crops, transforming goods, or being employed) or of one's wealth (giving your capitals in exchange for interest).
In the limit: yes, everyone can profit, since everyone has a chance to gain from an investment of it's own time - with or without society. You can be in a lonely island and profit from investing your time looking for fruits to eat. This profit might be bigger or smaller, but it will always be there.
The key question would then be the equality of these profits in society - not everyone has access to the same chances = not everyone gets the same profits.
Since you connected the question with money theory and inflation, your ability to profit will be balanced with growth and inflation rates, to make sure that the money reflects the real economy capacity. If you country discovers magic beans and everyone becomes rich, the goods in economy wont be able to satisfy the increase in wealth, so inflation will adjust the prices to your "new wealth", to match the supply of goods. Which leads us to the conclusion that more money for everyone would not solve inequality, would just create inflation.
I think the best way to answer you is to build a simple model. That's usually what economists do to answer such questions. I hope it will be useful.
Assume there is only one good in the economy, the 'consumption good', that people are all of the same generation or the same age and that one lives only for two periods (youth, and old age).
Then the answer to your question depends on the durability of this consumption good. How long can it remain useful to consumption after being produced ? Or, to think as an economist, is it expensive to keep the good useful ?
If it can for no cost remain useful for more than one period, then it can be rational for all workers to save simultaneously half of the monetary wage they receive as a compensation for their labor in period 1. Then at the end of period 1 firms will sell only half of the output produced by the workers. They will store the rest of it (we have assumed that this storage does not cost much). Then this stock of the consumption good will be available in period 2 and will be sold to the workers, who will buy it with their monetary savings. So the story is exactly the same as it would have been without money, if workers would have stored directly the commodities they produced.
If the consumption good is not durable, or is expensive to store, then this scheme is not possible. In reality, of course, many commodities of first necessity are not durable (goods but also services), and many durable goods are expensive to store. But that's not so much of a problem because we live in a world where generations do overlap, so the young produce for the old. When there is a demographic disequilibrium inside an aging country, then it should be able to trade with a 'younger' country, provided it has accumulated enough 'savings', i.e. valuable durable assets.
Edit. So far I only answered the question : is it possible for every one to live without working at the same time ? But the number of active workers can decrease with time if savings are used to invest in capital goods which make labor more productive.
It is perfectly possible. Consider a simple Overlapping Generation Model, with two generations, and young one ($y$), who work and consume, and an old one ($o$), who is retired and only consumes. After consuming, old agent dies, and a new generation is born. Population is constant. The savings interest rate is $r$. Each period, those who work earn an income of $W$.
The budget constraint of the young is
$$ W - C_{y} = S $$
where $S$ is savings.
The budget constraint of the old is
$$ C_{o} = S(1+r) $$
The lifetime budget constraint of an individual is
$$ C_{y} + \frac{C_{o}}{1+r} = W $$
(This is, present value of consumption equals present value of income)
Say the utility of the agent is given by
$$ U = C_{y} + C_{o} $$
Then, the optimisation problem the young agent faces when allocating lifetime income into consumption across periods (equivalent of deciding how much to save when young) is
\begin{equation*} \begin{aligned} & \underset{c_{y},c_{o}}{\text{max}} & & U \\ & \text{subject to} & & C_{y} + \frac{C_{o}}{1+r} = W \end{aligned} \end{equation*}
This gives the following optimal consumption levels:
$$ C^*_{y} = W\left(\frac{r^2+3r+1}{r^2+3r+2}\right) >0 $$ $$ C^*_{o} = \frac{W}{2+r} >0 $$
Conclusion
You have an infinite-period model with constant population, where both young and old are permanently having positive consumption. The young save to their retirement, which income they consume when old (together with the interest they received for their savings).
