I believe you can.
1. This year I gave away more than a doz. chicks. So chicks free, I even gave away 7 laying hens. Find this deal and your cost for birds is $0. One of those showed up at my brothers' friends house and I didn't want to keep her so I gave her away. Go to an auction and buy chicks or hens for cheap, an older hen can still produce viable hatching eggs.
2. Shop CL for free left over materials and build a coop and your cost for coop $0. I've seen free sheds for the hauling away. I buy culled lumber from Home Depot 2x4' peices of ply wood are big enough to build some nice nest boxes. 3 nest boxes $2-3 depending on the size.
3. Dispose of non producing hens and any non breeding roos. No room for pet birds if you want to break even, which I consider "supporting itself". I just gave away five roosters and two bantam chicks I didn't want.
4. Shop auctions and garage sales for feeding and watering containers. A wooden home built feed trough will do just as well as a store bought one. Tree branches on the side of the road from tree trimmers make great roosts.
5. Chickens don't care if there are "weeds" in the garden so growing extra greens for the birds is easy.
6. Sell, eggs, chicks, and manure. No one considers manure a cash crop but it can be to the home gardener. I once went to a plant sale where they were selling horse manure for $5 a bag. Don't laugh it sold. You can hatch successfully with broodies or a second hand incu.
7. Watch for feed sales. This year I bought feed at CM buy one get one. Stock up at these times. I repackage my feed to empy plastic cat litter containers to preserve it.
8. Trading is a way to figure $$$ from your chickens. I traded 4 chicks for goat milk soap and hay and goat cheese. These have value and should be considered income from your birds.
You just need to use your imagination. You also must consider entertainment from the birds has monetary value. Know what your customers want, colored eggs, brown or white. My SIL does not like brown eggs. (who knows) I also include educational information about the health of my eggs. A page from Grit magazine. Buy a copy of "Food Inc" and loan it out to family and friends.
Wishing you all the best
Rancher
forgot to mention one book said you can't with less than 10 hens.