The phrase "No Country for Old Men" comes from the opening line of a poem by W.B. Yeats called "Sailing to Byzantium."
The opening stanza reads like this:
That is No Country for Old Men. The young
in one another's arms, birds in the trees
--Those dying generations—at their song,
The salmon-falls, the mackerel-crowded seas,
Fish, flesh, or fowl, commend all summer long
Whatever is begotten, born, and dies.
Caught in that sensual music all neglect
Monuments of unaging intellect.
If you want a full analysis, you're going to have to head on over to our analysis of it, but we can do a quick summary here: basically, the line tells us that "that country" (which is unnamed) is a place for young and beautiful people who can still appreciate love and nature in all its passion. In contrast, the poem goes on to say, Byzantium is the right place for the old, somewhere they can turn their bodies (and their body of work) into aesthetic object. (Translation: they'll become works of art rather than bodies.)
Um, okay, so what does all this have to do with No Country for Old Men?
Well, tbh, we don't think the art bit has a lot to do with it. (We actually think that Cormac McCarthy might not have read much past the first two lines.) But those first lines are a perfect fit. We quickly figure out that the title refers to old man Sheriff Ed Tom Bell, who figures that he's no longer cut out to deal with all the horror and violence he encounters as a lawman.
As his even older friend Ellis puts it, "This country is hard on people." We find out that Ed Tom is pretty bored and aimless after he retires from law enforcement, but that's just the way it is. It may not be much of a country for old men, but it's the only one they've got. There is no Byzantium, and there's no changing the violence that's inside every one of us. All we can really do (in Ed Tom's words) is say, "Okay. I'll be part of this world."