There should be a health bar of sorts, two types. Systematic and Overall.
Systematic, how exactly? This game relies on you aiming your attacks, carefully striking, right? Depending on how 'deadly' you want combat I imagine this can be adjusted. But, for my example I'll use 'Super Deadly' as an example. Each Limb, the two arms, two legs, the torso, the head. Would have it's own 'Hp' bars, eventually they would reach a 'ruined state' where using them would be impossible. If not that route, than it could simply come down to 'taking' more damage if they finally reach 'ruined' and you keep taking hits.
What do I mean about the Limb health? Chidori. Let's use that. If I hit you in the chest full blast with one, assuming you have no chakra armor protection, etc. Do you expect to survive it, and be dashing around unharmed? Probably not, obviously overdoing it such as having working 'organs' so if they get you in a critical area is too much work.