It's not using the past because it's not that grammar. (If it were, I think it would probably be 体験したものではない , but I don't have much experience with that grammar).
The phrase そう体験できるものではない doesn't mean "impossible", it means "Not something you can easily experience."
そう ... ではない is a pattern with a meaning like "not too..." "not very..." ; "not very"+できる⇒unlikely, unusual.
もの is, I think, just an explanatory nominalizer.
There is also a grammar of 動詞+ものではない that is used to express what shouldn't be ; but I don't think that relates to this sentence.