We all know that Brook is the eccentric member of the Mugiwaras. He has a skeleton-like aspect because, when he was alive, he ate the Yomi Yomi no Mi, a Devil Fruit which allows its owner to revive after their first death; in fact, he really died in the end and his soul spent a whole year to find its body, so that it decomposed.
Despite this fact, the Devil Fruit had to accomplish its aim, so it made the soul entering Brook’s skeleton, making him revive like it was a normal body; in fact, in some moments we can see that, although Brook hasn’t a proper body made also of muscles and tendons, his skeleton can accomplish all the human functions, such as seeing (even if he has no eyes), eating and drinking (even if he has no stomach), speaking (even if he has no tongue) and so on.
Now, since nobody said anything about a hypothetical ability of Brook to recompose his bones once they have been broken, I think there are two main ways to kill him: first of all, he could be killed by destroying his whole body; we’ve already seen that, for example, if someone removes Brook’s head from the rest of his body, he will go on talking normally and he’s also able to hang it up again to his neck. Secondly, he can be killed also by using Big Mom’s powers for stealing him his whole soul, letting on the ground just a skeleton.
Apart from this two cases in which he would die for sure, in my opinion Brook is completely immortal, since his body doesn’t need food or water, he isn’t affected by the flowing of time so he can’t get older and, for this reason, he can’t reach an age in which he should be supposed to die.
“Ok, Brook is immortal, but if he’s weak, it doesn’t matter how long he could live for!”
Of curse, you’re right, but are you sure that he’s so weak?
Well, think about him from the moment he started fighting after the two-years time skip: we saw that Brook is now capable of using some new powers which come from the Yomi Yomi no Mi and that we’ve never seen before: in fact, for example, he can now leave his body in the form of a ghost (his soul) and freeze everything his sword touches thanks to the powers coming from the Reign Of The Deads.
These features are very important, because in Dressrosa Doflamingo said that, when a person awakens their Devil Fruit’s powers, they become able to unlock new powers and use them also to influence the surrounding environment; once we’ve remembered this, it seems obvious that, during the two years of training, Brook found the way to awake the Yomi Yomi no Mi and become a lot more powerful and skilled.