Can everyone in an economy profit? 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.
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.