"Physiological characteristics of bean seeds subjected to 25-year storage" is not a sentence. However, the writer has used a hyphen to combine 25 and year to create an adjective to modify storage. That's correct. What kind of storage? Twenty-five-year storage.
In that same way, "Seed Vault 25-Year Storage Life" would be better with a hyphen, because, again, the writer is combining two terms to create one adjective - this time, to modify Life. It's not a 25 life or a year life, it's a 25-year life.
So again, if you're using the two words to create a single adjective, a hyphen helps. Neither example is a possessive, so don't use and apostrophe or apostrophe-s.
As for "after seven years' storage," I've seen it both ways — with and without an apostrophe. The apostrophe means it's a possessive. Is it? It might depend on how you look at it. I'll let a grammar teacher answer that one. I can't.