If you don't already have some sort of structure for a coop, beware that your eggs are going to be quite expensive. I started with an existing garden shed that I converted into the coop and still wound up spending over $500 between "upgrades" to make the shed chicken-worthy, hardware cloth, welded wire fence, chicken wire (as a divider inside the coop), assorted posts and beams for the run, and fastening hardware. That would buy a lot of eggs. The chickens themselves were the least expensive part of the whole setup. I'm not trying to discourage you, just know that it's a valid argument your SWMBO could make. Granted I built a 16x16 run, big enough for way more than the 7 chickens I have. You might not need that much space.
The best way to convince my wife was actually having her hold the baby chicks and letting her fall in love with their fuzziness and "cheep cheep" sounds. She really came around after watching them play around in the brooder (before I'd actually spent any money on the coop/run) and especially watching them face-plant for a nap when they got sleepy.
All of that said, if I were doing this for my veggie garden I'd be comfortable with 2.5' wide at the base, and 2' tall in the center. That's big enough for birds to pass each other as well as stand up straight. I have Buff Orpingtons and Easter Eggers. For covering material, I'd personally either use deer netting or chicken wire as it is not a permanent home, just a barrier to keep them from digging up and eating my food. And since mine generally free range during the day, I can't really claim predator-worthiness as a consideration since for the rest of the time they aren't protected at all.
If you won't be watching over them, or they spend the rest of their time in a protected coop, you might consider something more sturdy in case a fox or neighbor dog wanders in, as neither of the above would stop a determined predator.
