First thing is first. Immortality does not mean you can not Die at all. There are two types of Immortality.
A Immortal body: This means that every cell in your body is Replaced at some point with new fresh cells and the old ones are cycled out this includes Bones and Teeth. Now the body will remain Alive as long as there is food, water, air and enough activity to keep it active indefinitely. This is one type of immortal body. This does not mean you can not die this just means that your body will never die of old age or decay because all the decaying cells are replaced at some point.
Physical Immortality: No type of Matter can destroy your body and your body can survive in any environment including the sun and your body is protected from Supper Gravity and radiation. This type is impossible for a person with a normal reality's technology and this requires High Level technology to supplement the human body with a Bio Mechanical suit which will maintain Body temps and Regulate Air into the lungs. These suits are capable of Allowing the skin to breath unlike normal clothes and utilizes a complex system to maintain a comfortable body while in the suit. Suits can take direct hits and internal organs are protected and Concussions are not possible Because of inertial dampening and Gravity Regulators.
This suit can absorb Kinetic, Thermal, Radiation and Electrical Energy.
Now onto the cool stuff. What kind of organs do you need to maintain a cycle of Fresh cells all across the body?
First thing is first, the current way of the body Healing external wounds is inefficient and does not Heal as soon as the wound is Created. There are lot's of way's for humans to die since we are so fragile. But first thing is first. We need to redesign the way skin operates. Skin is the organ on the body that continues to grow until you die of old age and even after you die it continues to grow a while after your dead. It's responsible for protecting your body from Germs and viruses and other harmful things.
Your skin Absorb's things and when it is cut sometimes things can get into peoples wounds that can result in death.
Redesigning the skin Is important Because it protects the body and is a major part of sex for humans. So maintaining a Fleshy feel while making skin resistant to environmental hazards and React to the skin being cut or pierced in any way.
Right now Your skin has sensors on it that tell's you if something is hot or cold and let's you feel an object when you touch it so you know your holding something. These sensors can be redesigned to send signals to surrounding muscles to clamp a wound and then they can be replaced at any time. With Ready stem cells. As you know stem cells can be used to regrow cells and can regrow entire organs. This cant just be in one place and there needs to be a way for these cells to track down the location to heal and start filling the cell damage with replacement cells.
Stem cells can Replace any cell you need. you just need to program them.