The answer to this question depends on which timeframe you are concerned with, and how optimistic you are about modern technology.
Finding copies of old, paywalled papers can be a problem. Nowadays, most papers are freely available on the web or at least in preprint form from the author's sites. DOI provides a robust way to link to them, even if the actual storage place should change. As the other answers state, you have probably no use for prints now (or in the near future).
However, what happens in the remote future? Once you stop caring -- maybe you switch career or (eventually) die -- the situation is similar with old papers today: readers are at the mercy of publishers. Is your paper still paywalled? Is the publisher still there? Has some search engine cached a version? Can PDF still be read on modern devices? In the worst case, your paper is practically inaccessible.
Does an archived paper copy help? Depends. There is no way any one place keeps hard copies of everything published. You can give your students and close colleagues hard copies for their own use, and maybe they keep so that maybe even in 50 years, an interested student who can not access your paper (easily) can get a copy from their professor that has undiluted value.
For example, Flajolet died. I am certain my boss (who works in closely related field) knew whom to ask for his academic remains. My boss has himself inherited all the paper accumulated and written by his late advisor (one of which I actually retrieved from the archive to check out for an answer on cs.SE; because it was impossible to find on the webs). This is stuff that does not exist on the web, but in real-world networks. For such, paper is important. Maybe that model is doomed given our technological advances, but I have the feeling that it will have its place for some time at least.
As far as I know, the issue of how we can keep our rapidly accumulating mass of data and knowledge at all and also accessible and organised over time is unsolved. It may be useful to keep that in mind.