When something sags, it leans, settles, or droops. If your bed sags in the middle, it can be hard to get a good night's sleep.
When you're tired, your whole body may sag, drooping with exhaustion. And older people are sometimes surprised to look in the mirror and see that their faces are beginning to sag with age. Objects sag too, with weight or age: "The porch on that old house has started to sag — the whole thing may need to be torn down." Sag is probably related to the Old Norse sokkva, "to sink."