Injured Animal Storage will come via upcoming upgrades to the Doctor's Office. We're assuming new goals will be released that must be completed before storing your fully healed animals (yes, they must be fully healed, just as other animals must be Adults before being stored), or at the very least you'll need to collect additional building materials to "build" the storage room in the Doctor's Office.
We've been told that other new features will be launched next week as well, but no other details have been given. You know us, though - the second something new launches in the game (whether it deals with Doctor's Office upgrades or not), we'll make sure you're the first to know (well, after we do).
Have you deleted any of your Injured Animals to regain space on your land? If not, are you happy about the thought of storing them, even if we'll likely have to work a bit to unlock the ability? Sound off in the comments.