This is a hard question to answer. First, notice that the question is about wealth and not money. The reason is because of the constant depreciation of money. We live in a world of fiat currencies where central banks are continually creating money out of thin air. So we could say that having one million dollars makes you wealthy, but that may not be true 10 years down the line.
Another similar question is how much wealth you need to retire. Again, it is hard to measure with money. It is not like you can just buy a bunch of 30 year bonds and live off the interest. If the dollar is devalued, then your fixed interest payments may not be enough. We also don't know what future interest rates will be and we can't accurately determine what rate of return we will get on our investments.
We could measure wealth in gold, but even that can be a problem. Because of the instability of fiat currencies, gold fluctuates wildly. You can have bubbles in gold, at least in terms of dollars and other fiat currencies. The price of gold went down in the 1980's while prices went up. The price of gold has gone up 5 times of what it was 10 years ago, but consumer prices have not risen that much during that period.
Gary North has written an article on debt. The particular part of this article that I want to point out is his comments on housing. He discusses someone who needs $5,000 a month before taxes in order to retire. He says, "I tell them that they need about six 3-bedroom, 2-bath houses that generate $1,000 a month net income before income taxes."
I think Gary North has hit the nail on the head with this one. It really is a good measure of wealth and a good measure of what you need to retire. If you own real estate that can be rented out, then you will be paid in dollars (or whatever money your country uses). Since you use dollars to buy consumer goods, this is a good measure. But it also solves the problem of inflation. As the dollar depreciates, rents will go up. Some of your other expenses like property taxes and maintenance may go up as well, but the increase in rent should far exceed this.
I am not saying that you need to buy real estate in order to retire (although I'm not discouraging it either). What I am saying is that it is a good measure of wealth and you can use it as a yard stick. If owning six houses, with no mortgage, will net you $1,000 a month and that is enough to live on, then you can calculate how much money you need right now. If such a house sells for $150,000 where you live, then you need approximately $900,000 (6 x 150,000). I would round it off to one million just to be safe.
But you have to make sure you invest it wisely to make it last, especially with inflation. To protect your wealth, I have two recommendations: real estate as mentioned above and Harry Browne's permanent portfolio plan. Both of these plans help protect you against a falling dollar.