Buy your lumber at your local yard or home improvement center. Plywood and pine are cheap and durable, but you can upgrade to more expensive woods that suit your taste.
Cut the lumber to the needed dimensions. These specs are for a full-size futon mattress. You can use them for other sizes of futon, but will need to adjust the lumber dimensions to suit.
Sand one face of each piece of lumber. If you want to paint or stain your frame, do it at this point.
Set one long beam and one short beam together in an L-shape. The end of the 50-inch beam should rest against the inside face of the 75-inch beam. Arrange them so the sanded faces of both beams are on the outside face of the L. Attach them with three of the four-inch wood screws. Set one screw in the center, and one each two inches from the end of the beams.
Repeat step three with the remaining beams.
Arrange the two L-shapes into a rectangle measuring 50 inches by 75 inches. Attach the two loose corners the same way to screwed together the L-shapes.
Lay the plywood sheet on top of the rectangular frame. It should fit so that its outer edges align with the outer edges of the frame. If it overlaps, trim it with your power saw.
Screw the sheet into place with one two-inch wood screw at each corner. Though it's not strictly necessary, you could reinforce the connection with a screw at the center point of each edge of the bed.