Feebleness is a state of being very weak. An old man's feebleness may make it difficult for him to care for his rowdy Great Dane.
In addition to physical feebleness, which people have when they're frail, fragile, or sickly, there's also a feebleness of will or character. The feebleness of your excuse for being late—because you couldn't find your favorite socks—won't impress your teacher. And the feebleness of laughing with a bully instead of defending the person being bullied may be the weakest of all.