Across The Universe takes place in an alternate universe to ours, affectionately referred to as Universe #1357.
This universe has quite a few differences to our own.
Earth is currently in the year 1998, however, modern technology is present and in full effect. Things like smart phones and flat screen TV's are already fully present. This is due to advances in technology and science, this Earth being 30 years farther technology-wise.
Earth is inhabited by humans, as well as anthropomorphic animals. Humans and animals such as canines, felines, etc. fully inhabit this Earth, and the whole universe for that matter, live together in harmony. Think; BoJack Horseman logic.
As well as other anthropromorphic animals, dragons exist as well; but they're a lot rarer. All dragons on this earth are from a large, floating island in the middle of the pacific ocean called 'Dragonslight Island'. Not a single dragon has ever left the island. Except for one.