I was really afraid of cats for some reason, especially when I was maybe 6 or 7. I don't even recall having a particularly bad experience with a cat that would've brought it on, but I recall it reaching its peak during my brother's paper route. Our mom would sometimes drive us around during the paper route (as opposed to us walking - it was a small town), and I recall this one street having only two houses to which we delivered papers. One of us would take the first house while the other continued to the other end of the street in the car with Mom and return to pick up whoever took the first house. I tried to avoid the first house like the plague, though I don't think I ever admitted to it, nor did I ever let on that it was because of a cat.
Anyway, one day, I got stuck with that house, and that particular subscriber always wanted their newspaper on their side porch. Knowing the cat could be there, I snuck up to the porch and climbed the steps to put the paper up there, thinking the coast was clear. It was then that the cat jumped up from behind the side porch and onto the porch itself. Not thinking I had any other options, I approached the cat and spoke to it to calm it (though it was probably perfectly calm already, honestly), then hit it with the paper and retreated back to the car with the newspaper still in hand. I don't really know what happened after that, so I'll fast-forward to the next time I got stuck delivering to that house...
It was wintertime, so I was wearing a big, puffy coat with a hood. When it came time to deliver to the house with the cat, I was afraid that it'd recognize me from the previous incident, so I pulled the drawstrings on my hood so the hood covered as much of my face as possible, leaving me with a small hole to look through. I didn't see the cat this time, thankfully, so the ordeal really didn't go any further.
Oddly, this is the only instance I recall of having to deal with a cat when I was deathly afraid of them.