There are two issues that must be untangled here:
1. No organism is indestructible
2. Senescence, the aging process
The first is trivial, while the second is the really interesting issue. Why organism-scale senescence? Why not perpetual good health?
Part of it is lack of indesctructibility on the cellular level; successful self-repair may require cells to commit to performing it, such as plant meristem and cambium cells and animal stem cells. Trees can live hundreds of years, but they are really thin growing layers that produce lots of dead material -- wood, bark, and dead leaves.
Another cause is lack of selection for survival after having successfully reproduced. Plants often "go to seed" and die soon after producing their seeds; a phenomenon also common in the animal kingdom. This can happen for two causes: lack of selection for continued survival and avoidance of competition with one's offspring. The latter mechanism is suggested by what looks like suicide mechanisms: male
Antechinus marsupial mice will die soon after copulating with their mates, having produced lots of stress hormones.
Also, inability to survive harsh conditions may provoke senescence; in temperate climates, many plants do not try to survive the winters; gardeners know them as annual plants. Their seeds can survive, and that's what keeps such plant species going.
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