I don't have a separate account or emergency funds just for my horses. I do have an account with an "emergency fund" but that is to cover anything and everything that we could possibly have go wrong (job loss, vehicle dying, medical emergency, etc..).
I personally have 1 years salary saved in our emergency fund. Hubby has been laid off 4x in the past 5 years, we have 2 small kids, a large dog and 2 horses, so I wanted to ensure that if we had a real "emergency" we would be ok. When unexpected bills arise, I try my best to to pay them without using our regular income by cutting back in other areas that month. If I cannot, my goal then becomes to repay my emergency savings as fast as possible.