While there do seem to be two workarounds for this bug, Zynga has marked this troublesome bug as "under investigation." Sure, it could still be a matter of days before the issue is fixed entirely, but at least you know Zynga has put the item towards the top of its to-do list, which is definitely better than never acknowledging it at all. Note: Some users aren't having any problems, while others (like myself - I'm so lucky), are still having tons of issues with the building.
How can you finish the related goals in the meantime? Two suggestions: Either refresh the game like man until it eventually works, or clear your browser's cache and try loading the game again. Both of these solutions are limited to a "your mileage may vary" guarantee, but they're worth a shot while you wait. Here's hoping this glitch is fixed sooner, rather than later, and we can start fully enjoying everything this new animal storage building has to offer.
Has your Big Barn performed a disappearing act on your Homestead? Have you been able to store animals in your Barn without issues? Let us know in the comments.