[myself]

We're keeping $2,000 in a high interest savings account as an EF. We'll be building that up to 6 months after we finish paying off our car loan.
Right now, if the emergency is too big, we use CCs (but budget around so that it's paid in full).
We don't consider a lot of things emergencies. Unplanned vehicle repair is an emergency, medical (other than a doctor's visit here/there) would be an emergency. There are probably a couple things more but I can't think of them off the top of my head.
We wouldn't spend any of the $2,000 just because there's a small leak in the house, we'd just eat a little less, and fix the problem ourselves.
As for heater breaking down, I'd say it's an emergency, but ours is only 7 years old, along with the whole house.