There are three common methods, two ballpark and one exact.
The first is two take a properly tensioned chain and try to pull it away from the sprocket. If you can pull the chain off the sprocket with just your fingers to the point of obvious daylight between the sprocket and chain, you have a worn chain. If it's just a small amount of pull, that means the chain will soon have to be replaced. If it pulls way off, you need to replace the chain. If you run your chain a little loose, as many really old bikes tend to run a tad loose, this method is not as reliable.
Second, as GT has said, you can lay out the chain on the work bench. Normally you take the worn chain, comparing to both a ruler and to a new chain. What you'll see when comparing the chains is that the pins/bushings don't quite line up. The amount of difference is a rough gauge of the wear in the chain. Then compare each to the ruler and you should be able to get a particular number for that difference. Pick a bushing/pin/rivet and line it up at the zero mark of your ruler. Count 24 pins/rivet/bushings (whatever you have opted to start with) and your last should be at the 12″ mark of your ruler (after all, a standard pitch chain is 1/2 inch). If it is off by much more than 1/16″ your chain should be replaced. Measure by 12 if you're using inch-pitch chain.
Finally, there are tools available to measure this. Park makes a gauge tool that you sink into the chain link holes to gauge the wear. Each gauge has a certain marking on it that will tell you the amount worn. It is actually fairly precise, but is also a tool only bike shops will have. Most people use method 1 or method 2.
Chain "stretch" is not "stretch" in the common sense - it's not like a rubber band being pulled. The chain's bearing surfaces on the pins/bushings/rivets are wearing away at each other, creating slop and play, this allows the chain to run long when pulled. It's not that the metal is somehow weakening, it's that there is a gradual wear-away of the bearing surface that introduces tiny additional space and slop into the chain joints.
Consider the chain a "wear" item if you ride a lot.