Only if you were very bad at financial planning. When you save up for retirement you're putting money away in investments such that you can live off the interest generated from your savings. Exactly how much you need depends on a number of factors like how exposed to risk your portfolio is and inflation rates, but typically a fund 20-25x your annual living expenses will allow you to live off the interest indefinitely. Putting that much away in 40 years is difficult and most people have to draw down their savings at the end, but with more time it gets much easier. For example you put away $300/mo for 40 years with 7% compounding interest you'll wind up with around 700k in retirement savings, which would give you about $35k/yr inflation adjusted in retirement - enough to live on but you'd probably have to draw down for things like vacations and emergency expenses. On the other hand saving the same amount over 80 years would give you $11.4 million inflation adjusted, enough to draw $570k/yr inflation adjusted forever. No one wants to work an 80 year career when life expectancy is around 80 years and the first and last quarter of that time you're not in a reasonable condition to work. But if you're living to be a few hundred years old, then those extra 40 years are nothing.

Of course this assumes that the economic system remains reasonably unchanged. It's possible that once we're all living such long lives and everyone has had enough time to build up large amounts of wealth that patterns of consumption and growth will be significantly different. But that said, the current expectation that people save up for retirement or die destitute is really a product of our current system and would likely change as well.