Like all woodworking, making band saw boxes takes several steps.
The first step is to decide on the wood you'd like to use. I have two ways I go about this.
The first is to think through the design and then choose the wood
that I think will go well. For example, lighter woods tend to look
smaller, while dark wood has a heavier look, as if it takes up more
I find that cherry is a friendly wood, while walnut and mahogany are more formal. Maple is a modern wood and oak is more "homey" and country. Of course that is a gross generalization, but I keep that in the back of my mind.
So, if I have a particular design that lends itself to a type of wood I'll use that wood, or some combination of woods and glue up a block the appropriate size.
My second method and one I use quite often is to go through my scrap pile and toss the likely looking chunks of wood to the side.
When I have an appropriate sized pile I'll sort through again and pull out what I like.
With this method I let the shape of the block of wood lead in the design, rather than determining the design first. This way I often end up making band saw boxes with random woods that I might not have considered putting together.
Once I've chosen the wood I'll glue it up into a slightly oversize block. When the block has dried I draw the design on the wood.
To make the box start by cutting 1/4" or so off the back of the block. This will be glued back into place once you've cut out the drawer openings.
Once you have cut off the back you are ready to cut out the drawer(s). Because you have already cut off the back you can now cut all the way through the block.
Notice that you will have to cut into the outside of the box to start
the cut. In this example, you can see two lines at the base where the
cut started and ended. You want to make these cuts with the grain so you
can glue the kerf back together.
You want to cut out the center in one piece because you will save the cutout for the drawer itself.
The drawer works the same as the band saw box itself, except you need to cut off both the front and back of the block.Then cut the shape of the drawer opening out of the middle section and glue the front and back into place again.
Once you've done it a couple times it makes more sense and comes more easily. I recommend playing around with scraps until you're comfortable with the process and with your saw.
Then let your imagination go wild. If you're not careful you'll be making bandsaw boxes until your shop overflows.