I really don't think "immortality" in the God-like never changing, impervious to all physical damage is what was being considered.  More like: not subject to natural aging.  If a tree falls on you, you still die.  That said, that is accurate as a description of the end of the universe.  It is a "problem" that is several billions of years in the future.  Even if immortality includes survival in the face of catastrophic accidents, I am willing to push off dealing with that problem for a billion years.  If anyone actually did survive for a billion years, I am willing to accept that boredom might be the biggest problem.  My earlier comment about always new opportunities or adventures really doesn't apply over  a billion years.  But over several hundred or even a thousand?  Bring it on please.  (If I am immortal enough to last for billions of years, I don't think starving or freezing will be my problem: the actual universe going away is not survivable - you are part of that universe).