Yes, you CAN mix your feed at home. But that said, it's not very easy or inexpensive, as people who ask this question often seem to assume. For most people with small backyard flocks, purchasing a balanced base feed is much, much easier than buying 5 or 10 separate ingredients in bulk quantities of hundreds of pounds and blending them yourself. Do you have space to store all that safely? Will you use it up before it spoils? Do you want to spend a bunch of time mixing feeds?
For most people, the answer is no. It's also a big responsibility. How confident are you that the recipe you're using is balanced for your flock?
That doesn't mean mixing chicken feed at home a bad idea; it's just something that takes considerable time and information, and it's something you should go into with your eyes open as to just how much responsibility you have, and what the consequences could be. Too much barley, for example, can cause tarry, sticky poo. Not enough protein will cause brittle feathers and a drop in laying. Not enough calcium will cause thin eggshells. Too little vitamin E and you might run into neurological issues, etc etc. Deficiencies and toxicities can both cause your birds to be less healthy and more vulnerable to illnesses.
The benefits of making your own feed can be quite nice, though, so if you want to go to the trouble, and you take the time to educate yourself, you can really customize your feeding to your flock's needs, and that can help ensure their health. For instance, you can increase the protein in your feed mix at molting time to help your flock during the annual molt. You can increase fat content in the winter to give them extra calories to help maintain their body temperatures during short days when it's cold and there is less time to eat before it gets dark. You can feed extra Omega-3s to help increase the Omegas in your eggs (foraging also helps this). The list goes on and on. And you avoid chicken feed recalls. They are not overly frequent, but they do happen--and sometimes for things like failing to provide adequate nutrition, like this Purina recall from 2014.
In the end, it's up to you. Some enjoy mixing their own feeds, and they do it well. Others just don't want the extra hassle or responsibility. If you're somewhere in the middle, you might consider purchasing a base feed, and making or offering some simple, natural nutritional supplements as treats--and as boosts when your flock may need it.
Of course, chickens with access to forage do some supplementation on their own, but if you have a situation where your birds don't get access to green pasture, you might consider supplements (pay attention to feeding instructions and offer in moderation), or growing Chicken Salad Seed in a bed raised bed for them.