Why doesn't capitalism run out of money? There's this question that doesn't let me sleep at night...

From what I hear, investing surplus money is always considered a good idea because it lets your money "work for you" and earn you more money, even while you do nothing (well, assuming you've made a thoughtful investment and don't just lose your money).

There are various ways to do this, but at the end of the day they are just a variation of the same theme: you buy some object O for X money and then rent it out to people for Y money per month (or week or year or whatever). Y is a fraction of X, but over time it amounts to way, way more than X. X can be anything from cars and boats to real estate and even money itself (aka lending). Companies are a special case of this where the owner rents the company to their employees. The rules are twisted here to be in favor of the employees (instead of a fixed Y they pay whatever surplus they produce; and if they don't produce a surplus then the owner has to pay THEM), but if a company is making a profit, that's essentially what happens anyway.

And the great deal about it is that you, the owner of the object, don't need to lift a single finger. You don't put in ANY work at all, you just rake in the cash. And then with the new cash you can then buy more O's and make money even faster and so on until infinity. Wealth this way expands exponentially, or at least it does until some real life misfortune happens to slow it down for a while.

But here's the problem with this that I cannot get over:

This means that the "owners" are like vacuums for money. There's a fixed amount of money in the system and they just keep getting more and more of it. Eventually, if nothing is done about it, you should end up with a few elite owners that own ALL the money, and vast masses of people who own absolutely nothing, because they've had to spend their last dollars to pay their rent payments.

The government can, of course, print more money, but that leads to inflation and we don't do that much anyway. You can tax the rich people and give a part of their income back to the poor, but that just slows things down.

So... why are capitalist economies thriving and average non-investing people keep getting better lives (and more purchasing power), when theoretically the opposite should happen?
First of all there are several incorrect statements in your text: 


Investing money is not just raking in the cash. It involves a risk, actually non trivial part of the reason why you earn interest on money you invest is the compensation for you taking this risk. If you consistently take bad risks you eventually lose your money, so its not correct to say investing is just like "rake in the cash". Moreover, if you invest in your own business/projects you have to put entrepreneurial work in. The only portion of the return that you could consider as "raking in cash" is the portion that you get to compensate you for opportunity cost of not investing the cash but enjoying it.
There is no economic theory that says in capitalism lives should get worse. In fact in economics we dont even use term capitalism - but assuming you mean by capitalism mixed but predominantly market based economies there are no economic theories which says life should get worse in those. Note in science we usually use theory differently from common English. A theory must be some consistent and testable explanation of some mechanism, and often we reserve the word theory only to consistent explanations that were already successfully tested and supported by evidence.
Money in economy circulates. Unless the rich investor stuffs the money under a mattress, it is incorrect to compare her or him to vacuum cleaner. If you put your money in bank it will get lend out to people who need it. If you spend your money on high living then again the money is paid to producers of the stuff you enjoy. 
You cannot invest any number of money you want and expect the same return. Ultimately you will experience diminishing marginal returns on your investment. As the supply of savings increases interest rate goes down as in a present situation when in some countries interest rate is even negative, so you get punished by saving and rewarded by borrowing money. 


Now to answer your main question, why the economy does not run out of money (assuming no more printing of new money), it is because prices in economy adjust. Generally, the monetary part of macroeconomic can be described by classical monetarist equation which states that:

$$MV=PY$$

Where M is the money supply, V velocity of money, P price level (change in which will give you inflation/deflation), and Y is the real output - how much goods and services economy can produce. 

Now lets assume that money supply is fixed (hence no more printing of additional money), and examine the effect of investment on this relation. First, real output actually depends on level of investment. The higher level of investment, the higher number of factories and capital workers can work with so also the higher output we get. So investment increases the real output Y. Real output is also actually what determines the well being of people within a country (at least as measured by lets say GDP per capita). However, to keep the equation balanced something else has to change. One possibility would be for velocity of money to increase- i.e. the speed in which money circulate, but velocity of money usually does not change much. So most likely what will actually happen will be that aggregate prices P decrease, meaning that for the same amount of money you will be able to buy more goods and services. 

Lets also for a second suppose that rich would actually stuff all their money in the mattress and never used them in any way possible. This would just be equivalent of actually decreasing money supply M and again P or V would adjust. In the short run decline in money supply could also negatively affect Y, but in the long run output Y is fully determined by the production capabilities of economy not by number of dollar bills circulating. 

So in either case sustained increase in material welfare is possible. Also, note redistribution is not actually about taking the money from rich and give them to poor its ultimately about transferring the resources from one to another. So redistribution also does not actually necessarily depend on availability of money in the economy (although thats a topic of its own that I wont go into).

So to sum it up, some of the premises in your question are not correct, and the answer why you can have a sustained increase in wellfare when money stock declines is that prices (or potentially also partially velocity) adjust. The total output and hence material welfare of country is (in the long run) independent on quantity of money.