It is very much possible to have backups. True. 

But the important thing here is, the moment you die inside The Matrix, your brain basically shuts down it's signals to the lungs, heart etc, which kills you physically. And once the lungs and the heart shuts down, the brain does too!

So even if the person's mind is backed up in The Matrix, there is no way to make the brain restart the lungs and heart!

An important point to note here is that The Matrix only generates sensory inputs. 
Think of it like using your keyboard to delete your hard-disk. This will render your machine un-bootable! Similarly, the sensory input of, say being shot at(Formatting your hard disk), will cause the brain to shut down your body(rendering it un-bootable).

The biggest proof for this is seen in Reloaded/Revolutions, where Smith has overwritten Bane(A red pill). This itself proves that

1. Personalities can be read and written from within The Matrix to the real world & back.
2. Though regular Matrix broadcasts consist of sensory input, the same channel can be used to transfer entire personalities.
3. Personality downloaded from within the Matrix can be retained even after being disconnected from the Matrix.