This can be pretty intimidating if you are not familiar with building, but it is something most people can do. One of your problems may be that you don’t have the tools you need or the experience using them. It can also be time-consuming. But first you need a plan.
Chickens’ needs really are pretty simple. In Connecticut they need protection from the worst of the environment, protection from predators, a roost to sleep on, nests, a way to get in and out to the run, and a place for food and water. The chickens really don’t care what it looks like but in suburbia especially you probably want it to look decent so your neighbors don’t complain. You can get a lot of different opinions on how big the coop needs to be. You can get some of my thoughts by following the link in my signature, but that is just opinion like everyone else’s. I find I have fewer behavioral problems to deal with, more flexibility to deal with any problems that do pop up, and I have to work less hard if I give them a bit of extra room. In suburbia and with limited space, poop management could become an issue. That’s probably where you put in extra work if you crowd them. And room for them does not just men coop space. It means how much room is available to them when they are awake. That could be coop, coop and run, or some other combination. So instead of thinking about coop space in isolation, think about all your facility.
For a flock of no more than four hens in your area, all hens about the same age, I’d look at a 4’x4’ or 4’x6’ coop with a run somewhere around 4’x8’ as a minimum. Personally I like a walk-in coop but that gets big and expensive. You probably want your coop section elevated a couple of feet so the chickens can seek shelter underneath it.
You can look in the coops section up at the top of this page. Many of those come with plans. Another option is to go talk to your librarian to look for shed plans. Those books often come with construction tips and techniques. A Lowe’s or Home Depot probably have books that will tell you how to build a shed, pictures showing construction techniques and all that though your librarian should be less expensive than buying that book. A coop is just a shed with a roost, nests, and a way for the chickens to go in and out. You can modify the shed to suit though you would also have to come up with a run construction plan and maybe a way to support the elevated shed.
You could attempt to build it yourself or you could look for a handyman to build it for you. How do you find a handyman? That can be a challenge to get one that is dependable. You can ask neighbors and friends, people at church, Angie’s List, or chat with the people at the building supply store. A real estate agent can be a good source if you have contact with one of those. The potential problems with asking anyone is that it may be a relative that needs a job, someone they have heard about but not used themselves, or their standards may be quite a bit different than yours. It’s a chance you take.
It’s hard to get all that done with new material plus paying someone to do it for $300 though. Real hard. Another option is to look for used materials, maybe on Craigslist or you may have one of those “habitat” places that recycle construction materials. When looking at expenses, don’t forget the hardware. Hinges and latches can add up in costs pretty quickly. Sometimes you can get some pretty good material at a construction site. They often use some pretty nice stuff as construction aids and just throw it away after they have used it.
Another option is to try to find a building on Craigslist or somewhere that someone wants to get rid of. Maybe you can just move the building and modify it to make a shed into a coop. You may need to take the building apart just for the materials. Sometimes you can get lucky like this. Old dog kennels can get you some decent fencing for your run.
Your $300 budget is going to be rough but there are many different ways to go about it. You can do it but it would probably involve scavenging much of the materials and doing most of the work yourself. $300 doesn’t get you much these days.
Good luck!
And where are my manners? Welcome to the forum!