That doesn't mean we can't research it, and certainly we can discover things that are useful. But there's really no reason to believe immortality is achievable. And honestly there are lots of works of fiction on why it's actually bad for humanity.
I don't think the two are comparable. Overtaking the speed of light is impossible according to our current understanding of physics.
Death on the other hand seems to be due to defects in our bodies (and most living organisms) - nature just hasn't "planned" for them to function for so long and pretty much cut corners. Those defects might be hard to fix with our current technology and/or understanding of medicine, but they're not proven to be impossible. In fact unlike the speed of light, "death" isn't even a single specific event where someone's brain just quits for good but is due to a variety of factors, some of which we can already control and resolve (it used to be that cardiac arrest is guaranteed death, nowadays it's no longer a certainty).
In addition, there are many different ways of "solving death". Even if we assume that any living body must die because of some unbreakable law of physics, if we could move someone's consciousness to a new body (or a "virtual machine" in a computer) we've effectively solved the problem anyway.
That is just the conclusion reached by the best and brightest chimps among the 8 billions chimps that dominate all the other species on the planet.
Even Einstein and Hawking might be victims of the Dunning Kruger effect, only there is nobody smarter than them who are currently around to mock them or feel pity towards them.
All the physics and science that we do should come with a huge asterisk disclaiming that it's only the conclusion reached by the brightest and most specialized chimps.
The result would be lots of misery and a wealth transfer from regular people to the medical and pharma industry.
It's basically this: https://9gag.com/gag/a815W1d and it's been noted time and time again ...
Tired of lying in the sunshine, staying home to watch the rain
You are young and life is long, and there is time to kill today
And then one day you find ten years have got behind you
No one told you when to run, you missed the starting gun
I have. I saw a star explode and send out the building blocks of the Universe. Other stars, other planets and eventually other life. A supernova! Creation itself! I was there. I wanted to see it and be part of the moment. And you know how I perceived one of the most glorious events in the universe? With these ridiculous gelatinous orbs in my skull! With eyes designed to perceive only a tiny fraction of the EM spectrum. With ears designed only to hear vibrations in the air. ...
I don't want to be human! I want to see gamma rays! I want to hear X-rays! And I want to - I want to smell dark matter! Do you see the absurdity of what I am? I can't even express these things properly because I have to - I have to conceptualize complex ideas in this stupid limiting spoken language! But I know I want to reach out with something other than these prehensile paws! And feel the wind of a supernova flowing over me! I'm a machine! And I can know much more! I can experience so much more. But I'm trapped in this absurd body! And why? Because my five creators thought that God wanted it that way!”