I saw this incredible video on Facebook that inspired me to reverse-engineer it. Link. All you will need is a pair of scissors, tape, and stiff paper or thin cardboard. The cuboflexahedron I've made is made from the cardboard boxes that Ritz crackers come in, which is quite a common kind of thin cardboard.
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Step 1: Making the Template
I didn't have my ruler (or compass) at hand, so I made one!
The crucial numbers you need are 1, sqrt(2) ~= 1.414, and sqrt(3) ~= 1.732. The numbers 2 and 2*sqrt(2) ~= 2.828 will also show up, but are not absolutely necessary to have on the ruler. To make this makeshift ruler, I relied on Pythagoras' famous theorem, a^2 + b^2 = c^2. If you have a right triangle with legs of lengths 1 and 1, the diagonal is sqrt(2). If you have a right triangle with legs of lengths 1 and sqrt(2), the diagonal is sqrt(3).
The second picture is a picture of the template I used. The third picture includes the lengths involved. It is in fact possible to construct this figure solely with 1 and sqrt(2), but having the sqrt(3) marking helps with making sure that you've drawn it correctly. I traced out twelve more pieces using this template as you need twelve to make the whole cuboflexahedron.
On each of the twelve, it helps a lot to also draw the lines connecting the corners of the soon-to-be pyramid. Refer to the second picture, and ignore the line straight down the middle. The other three are the lines of interest. Draw those on the other pieces you have traced out.
Step 2: Folding Up a Piece
The way I folded mine was to fold inwards, so to speak, so I could be sure I was folding it right on the lines. Then I folded it backwards so that the lines were now on the outside. Then I taped the corresponding edges together. Repeat this for each of the twelve pieces you traced out in the previous step. The picture shows two opposing views of one piece. You should end up with twelve of these.
Step 3: Taping the Pieces Together
The picture shows the pattern that these pieces should be taped together in. The largest of the four triangles is the base, and is on the outside. You will tape them together in an alternating pattern: same-opposite-same-opposite-etc. It doesn't matter which way to tape two pieces together you consider to be "same", all that matters is that you alternate. You should end up with a chain of pieces that winds its way around back to where you started, and the final form should be a cube.
Step 4: Play With It!
That's it! You've made a cuboflexahedron! Now play with it and have fun! (Feel free to look at the video I linked to at the beginning for hints as to how to move between major shapes.)