Does Science Say That We Should Die?
Quoting a great biologist Peter Medawar :


  It is by no means difficult to imagine a genetic endowment which can
  favor young animals only at the expense of their elders; or rather at
  their own expense when they, themselves, grow old. A gene or
  combination of genes that promotes this state of affairs will, under
  certain numerically definable conditions, spread throughout a
  population simply because the younger animals it favors have, as a
  group, a relatively large contribution to make to the ancestry of the
  future population.


From Why We Get Sick :


  Imagine that there is a gene that changes calcium metabolism so that
  bone heals faster, but the same gene also causes slow and steady
  calcium deposition in the arteries. Such a gene might well be selected
  for, because many individuals will benefit from its advantages in
  youth, while few will live long enough to experience the disadvantage
  of arterial disease in old age. Even if the gene caused everyone to
  die by age 100, it would still spread if it offered even minor
  benefits in youth. This argument does not depend on the prior
  existence of senescence. Other causes of death—accidents, pneumonia,
  and all the rest—are sufficient to reduce the population at older
  ages. Nor does the theory depend, like Haldane’s, on cessation of
  reproduction.


The message I am trying to get across is that - As organisms are like vehicles which carry DNA and DNA just "madly" wants to replicate itself and pass itself from generation to generation, Natural Selection will favour those traits that maximise reproductive success. These may be traits that provide maximum benefit in youth but cause gradual damage and lead to senescence and death. 

You could also think like this - Every machine has wear and tear and must be thrown away after it is irreparable. Same is with our body, it is the most beautiful machine in the world but it is also subject to the same wear and tear.

Also there is the end replication problem. See telomeres if you dont know what I am talking about. There is a limit to number of times a cell can divide. 

So all in all I would say that there is a biological limit - a certain age up to which a person may live.