You need to research and activate them in your research tree. If you have a blue icon next to it, you have to put more knowledge point to it. If you have a yellow icon with a dollar sign, you have to pay some resources to activate it. And if you have a green tick next to it, you already activated this research.
If you have a green tick and you still can not train archers in your barracks, then it can be a bug, in this case you should contact the in-game support, as they can see more details on your progress and help you more efficiently in your case.