Oh hey, there was another layer of fabric under there!
I made sure to not vacuum before I put the fabric face down on the ground. That way the fabric would get nice and coated with dog hair.
I cut out a piece of fabric and started stapling it to the chair bottom. I found this staple gun when I bought the house in a kitchen drawer, along with the aforementioned hammer, and a wrench. I love free tools.
Then I trimmed the excess fabric away and screwed the seat back onto the chair frame.
It looks so much better, yes?
Then I started on the rest of the chairs. And promptly ran out of staples. Damn it.