You can set up chickens to only be cared for once a week pretty easily. I think it's actually pretty easy to build a predator proof coop and run. It's just expensive. Sick dogs and cats can stay at the vet and so could a sick or injured chicken, if you make arrangements.
I agree that it would be important to have a couple of dependable people in the area, that you could count on. Ideally, someone that would like to collect eggs and just make sure everything was ok. At the very least, someone you could call if an emergency came up and you couldn't get there that week or the weather looked bad. If I couldn't get someone to check on them regularly, I, personally, would set up a web cam, so I could see that everything was fine in the coop.
Once you got attached to them, you might like being able to watch them and listen to all their little noises during the week, when you're back in the city.
Would I do this? No, but I have a totally different lifestyle. I've made different choices in where I want to live and how I want to live. Some people put more time into their chickens than others and some people put more money into their chickens than others. A lot of people on this forum make different choices than I do about how they care for their chickens and how they deal with predators. To me, this is just a more extreme version.
If you're willing to invest the money and effort into creating an environment that is safe and meets their needs, I say, go for it. Keeping chickens doesn't have to be expensive, but with an absentee owner, I think you'll have to make a greater investment to ensure their safety. You may decide the extra effort and money is worth it or you may not. You'll just have to think about it and make your own decision.
