Assuming you don't have a small house, in the main room (the one with the large dining room table), on the top floor in the back left corner is a crafting station. You can make two small beds and one double bed. The small beds are for kids.
In addition to beds, though, you also need a container of some kind (I'm assuming it's a dresser, but I've just always decked out the full house).
But as long as you have a free single-space bed and a container with it, you can adopt a kid.
You do not need the west (or east, I forget) wing bedroom to have children in your house, nor to have your own bed (you'll get the double-sized bed automatically).
Otherwise the dialogue option (from what I've read) is something to the effect of "I want to adopt you, but I don't have room right now."