Q:

How much lump sum investment in stocks would be needed to yield a target stable monthly income?

Situation: a guy has a lot of money. He wants to invest in stable and low risk stocks so that he could get a steady 1000$ a month income at least for the next 10 years (and having equal or bigger repository of stocks afterwards.)
Location: USA / UK / Western Europe. How much does he have to invest? Where? In the stocks of huge companies like Google, or packages of stocks? The guy doesn't know a thing about economics.
P.S. I'm that guy, only without money. Blind guess: 500,000$.
Short answer: 300,000$ and up, with the risk going down as the price goes up.

A:

If your requirements are hard (must have $1000/month, must have the same or bigger in capital at the end), stocks are a poor choice of investment. However, in many cases, people are willing to tolerate some level of risk to achieve the expected returns.
You also do not mention inflation, which can take quite a lot out of your portfolio over the course of ten years.
If we make some simplifying assumptions, you want to generate $12,000 a year. You can realistically expect the (whole) stock market, long term (i.e. over time periods substantially longer than 10 years), to return approximately 4 - 5% after factoring in inflation. That means an investment of $240,000 - $300,000 (the math is simplified somewhat here). If you don't care about inflation, you can up the percentage rather somewhat. According to this article, the S&P 500 returned an average of 11.31% from 1928 through 2010 (not factoring in inflation), which would require an investment of approximately $106,100.
But! This opens you up to substantial risk. The stock market may go down 30% this year! According to the above article, the S&P returned only 3.54% from 2001 to 2010. Long-term, it goes up, but your investment case is really unsuited to investing in an index to the entire stock market given your requirements.
You may be better suited investing primarily in stable bonds, or perhaps a mix of bonds and stocks. Alternatively, you may want to consider even more stable investments such as treasury notes. Treasury notes are all but guaranteed, but with a lousy rate of return. Heck, you could consider a GIC (that may be Canada-only) or even a savings account. There's also the possibility of purchasing an annuity, though almost everyone will advise against such.
Personally, I'd go for a mutual fund which invested approximately 70% bonds and the rest in stocks over such a time period. Something like ING Direct's Streetwise Balanced Income Portfolio, if you were in Canada. It substantially lowers your expected return but also lowers your risk. I can't honestly say what the expected return there is; at this point, it's returned 4% per year (before inflation), but has been around only since the beginning of 2008. And to be clear, this is absolutely not free of risk.

A:

I will add another point to ChrisinEdmonton's answer... I recognize that this is perhaps appropriate as a comment--or maybe 1/2 of an answer, but the comment formatting is inadequate for what I want to say. 
The magic formula that you need to understand is this: 

(Capital Invested) * (Rate of Return) = (Income per Period)

When ChrisinEdmonton says that you need $300,000, he is doing some basic algebra...

(Capital Required) = (Income per Period) / (Rate of Return) 

So if you're looking at $12,000 per year in passive income as a goal, and you can find a "safe" 4% yield, then what ChrisinEdmonton did is: 

$12,000 / 0.04 = $300,000

You can use this to play around with different rates of return and see what investment options you can find to purchase. Investment categories like REITs will risk your principal a little more, but have some of the highest dividend yields of around 8%--12%. You would need $100,000--$150,000 at those yields. 
Some of the safest approaches would be bonds or industrial stocks that pay dividends. Bonds exist around 3%--4%, and industrial dividend stocks (think GE or UTX or Coca Cola) tend to pay more like 2%-3%. 
The key point I'm trying to make is that if you're looking for this type of passive income, I recommend that you don't plan on the income coming from gains to the investment... This was something that ChrisinEdmonton wasn't entirely clear about. It can be complicated and expensive to whittle away at a portfolio and spend it along the way. 

