To save on cash, most woodworkers will fashion the base of their box out of plywood or maple for strength, and cover it with an ornamental wood as a veneer. An alternative would be to craft the inner part of the box out of the lower quality wood and applying the trim parts and lid in a solid wood of the desired type.
Baltic Birch plywood is one of the types of plywood that is desired because it's face is pretty and it's very strong and durable, being many more plies than typical plywood.
You should be able to find some veneers at a local lumberyard or through special order at one of the big box stores. Alternatively, you can look around for a local cabinetry shop or lumber yard and either ask for off cuts or some rough stock to work with in a species that they have on hand or can order for you. Common woods you should be able to get are: Cherry, Mahogany, Red Oak, White Oak, American (Black) Walnut, Pecan, Poplar, and Maple.
Once you have the size of the drawers measured out, you can cut those drawer bases to size, and give them a base coat of paint (in case the felt rips later). And then I suggest you take some spray-on glue (3M or what have you) and adhere a sheet of felt to the front, pressing it into all the various nooks really good. Then go around it with a sharp pair of shears to cut away the excess, leaving about a 2mm overhang or less all the way across the top and bottom, and wrap the excess material down the sides for best appearance. The 2mm excess will cover any gap between top and bottom of the drawer, and the wraparound on the sides will do the same for all the edges.
Gluing the drawer bottoms down when you're finished might be a good idea, or you can cover the bottoms with felt as well and add little loop handles at either side to allow them to be removable.
