Stack two sheets of newspaper on top of each other. This will be a two-ply bag for extra sturdiness.
Cut out a rectangle that's 15.5" wide and 8.25" tall. If your paper already has a fold in it, align the existing fold with one of the fold lines in the diagram below, unless you don't mind an extra fold appearing somewhere on your bag. I cut out this rectangle, then flipped the paper over so the blue area would be on the outside of the bag.
Fold a flap 1.25" down from the top. Fold a flap 2" up from the bottom. Then measure off and make vertical folds in the places shown in the diagram above. The front and back panels are 4.5" wide, the side gussets are 3" wide, and you'll need a 0.5" flap for gluing the bag together.
Cut two pieces of cardstock or chipboard to 4.25" x 1", then glue them on the widest two panels just under the top fold. These will reinforce the rim of the bag. Glue the top flap down along the length of the bag, covering the cardstock. Since the bag is two-ply, you'll need to glue both flap pieces down one at a time.
Put glue on the outside of the 0.5" tab and bring the left-most panel over to form the body of the bag, aligning the cut edge of the panel with the folded edge of the flap. Add a little more glue to make sure the outermost sheet of newsprint gets tacked down, too.
Upend the bag so the 2" flap is now up. Fold the short sides inward as if you were wrapping a present. If it seems easier, you could also lay the bag on its side and crease those folds against the table.
Put glue on both flaps and fold them inward to form the bottom of the bag. Standing the bag upright and pressing down from the inside will help to secure them.
Cut a piece of chipboard to 4" x 2.5" and glue it to the bottom of the bag to reinforce it and hide the flaps if you want to be an overachiever.
Punch holes in the rim of the bag, adding eyelets if you like, and string some cord through the holes to form handles. Knot each end of the cord so it won't pull out through the holes.
If you want to store your bag flat, pinch the top together, fold in the sides, and bring the bottom up so that it lies flat.