This is less a quirk of DNA and more a quirk of how we define things. You could as easily ask if proteins are alive, or the carbon, hydrogen, oxygen and nitrogen that comprise you. We have a linguistic understanding of reality which demands that we draw these boundaries somewhere to give each word a constrained meaning, but forget that we're the ones drawing the boundaries. We cut reality up into smaller pieces so we can more easily interface with it, then puzzle and marvel at how it all fits back together. But any carpenter can tell you that every time you make a cut, something is lost. If you cut an eight foot board in two, the pieces won't add up to eight feet anymore, they'll add up to eight feet minus the width of the blade. The material in between is turned to dust. The more divisions you make, the more of the whole you lose in the process. This is necessary, of course; you can't do much with a tree while leaving it a tree. But you also can't put every grain of sawdust back in place.
For the retards in the audience: reality is the tree, words and concepts like "alive" are boards we cut from the tree to make it useful, defining those words and concepts is the act of cutting, and question like OP are sawdust.
