This graphic (PDF warning) --
-- shows that a person's lifetime (say, 75 years) probability of dying of any of the top four non-biological causes of death (suicide, car crash, toxic exposure, falling) is 0.033 (roughly 1 in 30). Let's say this is for a normal biologically limited lifetime of 75 years. Using this as a basis, the Binomial Theorem shows that we have a probability of .000448 per year of dying of any one of these causes, and, for 75 years, this probability produces the quoted chance of 0.033 for any of these four causes. For longer lifetimes, this outcome means that:
My Python code for this result:
# -*- coding: utf-8 -*-
from scipy.stats import *
k = 0
p = .000448
print(' %8s %12s' % ('Age','Probability'))
print(' %s' % ('-' * 24))
for n in (75,100,200,500,1000,2000,5000):
r = 1 - binom.pmf(k,n,p)
print(' %8.0f %12.4f' % (n,r))
And this is without even considering other technological advances that would increase safety during the time from 75-5000? Heck, we even have the prospect of self-driving cars now which would significantly decrease the chances of dying in a car crash. I think this only shows how robust the indefinite lifespan theory might be.
To a certain degree one has to come to terms with the fact that most people in the world hold incoherent views on the topic of aging to death. If asked most of them say they want to: they want to do exactly the same thing they see other people doing. Conformity is a hell of a drug. But if asked if they want to suffer heart disease or Alzheimer's, they don't want that.
Then there is the hairshirt environmentalist death cult that tells us we're destroying the world, there are too many people, and we should all die. That is not actually much of an exaggeration of the actual viewpoints espoused. This has so pervaded society that you can find the average person on the street is concerned about living too long because they are someone burning up a limit faster. It's ridiculous. People look at the horrible consequences of war and kleptocracy in various regions , and buy into the propaganda that this would somehow go away if fewer people were involved: that it isn't the mass murder, or the oppression, or the theft of resources, or the enforced poverty, but just a matter of how many people there are. They fuse this into the desire to confirm to the lives of their parents, and justify the continued lemming-like walk towards the suffering and pain and disease of aging.
But if you ask them if we should cure any of those diseases, they're all for that. They just don't want to live longer. But this is the greatest incoherence: age-related diseases are age-related diseases because they are caused by the underlying processes of aging. The only true cure is to repair the damage of aging  that causes these diseases. As for all machines, if you fix the damage it lasts longer in good condition . Extension of healthy life and true cures for age-related disease are flip sides of the same coin.
And if treatments for aging were widespread and available, all those people who say they want to die on the same schedule as their parents would use them, and live longer, and get over it. But without their support the necessary research will not run fast enough to save most of them. They'll die for their wrong ideas, right on the eve of the transition to a world of indefinite life spans achieved through medical science that can control the causes of aging.
If I recall, some varieties of jellyfish and lobster have this.
A number of lower animals seem to have indefinite life spans, but that may be because few people are willing to put in the time to figure out whether they only have very, very long life spans. Hydra, for example, are one of those cases.  These species are not terribly relevant to any human medicine. Hydra are (probably) long-lived because they are ferociously competent regenerators, little packages of stem cells essentially. Once you have a complex nervous system you can't run the "regenerate everything all the time" model, however, as the fine structure of your brain needs to stay the way it is.
The AnAge Database lists a few more species considered to have negligible senescence that are of more interest because they are more complex organisms.  Longevity figures are often a question mark here, as there is too little study to determine what life spans actually are if these animals are left alone in a place where they have food and won't get eaten or diseased.