This cardboard barn can be built with only 3 things - cardboard, razor blade and tape (though more tools are recommended). As for ingredients beyond cardboard and tape, I suggest a 3rd magic ingredient of your choice - such as LED lights or a light coat of spray paint.

The construction is scalable (see diagram). Once you define your unit and make a "jig" (your special ruler), you don't need a ruler. You actually don't need a ruler at all, you can just define some arbitrary length (my child is about knee high, and I want the door to be over her head by a quarter knee, etc.) and use that as your length. So if you have enough cardboard, your barn can be 5.5 feet high or 5.5 fingers high. The geometry works out the same.

The structure can be built with only a razor blade - using razor blade nicks in the cardboard instead of a marker or small pieces of tape (cutting the tape with a razor blade instead of scissors). And there's a place where a string is handy, but an long narrow oval made of cardboard or a length of tape can substitute for a string to draw the roof and door arch. Again, a ruler is handy for calculating center lines and maximizing your cardboard, but marking a piece of tape (sticky side in) and folding it in half will serve almost as well to find the midpoint as seen in the steps below.

My example was built in an afternoon from a pile of cardboard we already had lying around.

For clarity, the unit of measure for this example was derived from the height of my largest piece of cardboard (39.5 inches) divided by 5.5 to give 7 and 3/16 inches. The depth of the barn (the length of the boards) does not matter. I went with the length of one of my shorter boxes to be sure I had enough cardboard for the 14 "boards."

Let's begin!

Step 1: Preparing Tools

Essential items:

-Cardboard (the cardboard also supplies a straight edge)

-Razor Blade or Box Cutter

-Tape

I suggest a razor blade for safety and control. I suggest packing tape over duct tape (gasp!) simply because I know my wife won't want our child around the smell (chemicals!). Of course duct tape would work better, so use what works for your situation.

Step 2: The Jig

The jig is the key to a well built barn.

Use a straight edge that comes with one of your boxes to make the jig. If you're actually going to build everything using razor nicks instead of a marker, make your marks with slight pressure then go back and cut (measure twice, cut once).

The jig should have center markings for one unit and side markings for 1/2 unit on each side (and a middle mark). The walls will be measured using whole units, but the boards need the half units. If you are considering dovetail joints (not explained in detail here since that wasn't what I did), I suggest you add 1/4 unit markings inside your half units for making 1/4 slots instead of 1/2 slots.

If you're going the razor-blade-only route, cut a piece of tape one unit long and fold it in half for the 1/2 unit. This will be the general method for finding the center/middle of pieces of cardboard if you don't have a ruler or tape measure.

Step 3: Building the Walls

Building the walls is pretty well laid out in the pictures.

Use the jig and a second box as a T-square and straight edge, find two pieces of cardboard that will work for your 5 x 5.5 unit walls.
Mark the base and sides using the jig and a straight edge. Note - it's easier to cut along a continuous line than a dashed line. Go ahead and mark the cuts as well as possible.
Draw the top half-circle. Use a piece of tape or string to find the middle of your wall. Use the jig to measure 3 units up the center of your wall. Use cardboard, tape, or string to keep your razor (or marker) to center distance a constant length (2.5 units). Make sure your arc will intersect the side walls before drawing it in. Adjust your "compass" length as needed. Note - using the centering method will generally give better results than just the jig, but middle line locations should be confirmed with the jig when possible.
Use the jig to mark the slots for the planks (whole units). Use the 1/2 (or 1/4 for dovetail) unit marks to set the slot depth.
The bottom slot is tilted at a 45 degree angle. This enables a secure base with the bottom plank. Use a piece of paper folded at the corner (or the same with a piece of tape) to find 45 degrees. Again the 1/2 unit depth is marked using the jig.
I show scissors in the picture, but every cut I made was easier and cleaner with the razor than with scissors.
Remove all excess material. Cut the door last (see Assembly Preparation).

Step 4: Laying Planks (Building the Boards)

The side boards are interesting to make, but are the most labor intensive part of the construction - particularly the pre-folding.

Determine how long (deep) you want the barn to be. Make sure you have enough cardboard to make 14 boards. If you have plenty of cardboard, you could just cut 14 pieces and then make them all the same length later based on how much floor space you have.
Plan ahead tip - already existing folds near to where you will make your plank folds makes creating the new fold very difficult. (Try to fold a piece of paper in half, then try to make another fold right next to the first fold to get an idea. The second fold tends to "slide" into the first fold.) I only had one plank with this issue. But it's difficult enough to consider including more lid-cardboard to avoid a pre-existing fold.
For each piece of cardboard to be made into a plank, find the middle of the sheet of cardboard using the folding tape/string technique (or measure with ruler/tape measure). Mark this line to use to align the center of the jig with the center of the board. In some cases I had cardboard large enough to give two planks and I cut that in half first then drew the middle lines in.
Use the jig to mark the four lines - 2 outer cut lines (2 units apart) and the 2 inner lines (one unit apart). The 1/2 unit between the inner and outer line is the part of the board that will fold in to go into the wall slots.
Cut the board out then tape any gaps resulting from lids after you cut out the sides of the board.
You now have 14 flat boards. Folding time! If you don't have a thin board or piece of plastic, a sharp table edge might work. Also nicking the outside of the fold with the razor will help make the fold. It turns out you get better structural integrity from clean folds than you lose from partially cut boards.
You now have your parts created. Two walls and 14 boards.

Step 5: Assembly Preparation (and Doors)

The planks will extend beyond the front and back walls by about 1/4 to 1/2 unit. I suggest marking this (or dovetailing, as mentioned before) to help keep your planks even while working inside or outside the structure. My wife helped hold the walls up as best she could while I taped everything together.

But first - the front doors:

I suggest cutting the doors out last to avoid any accidental tearing or folding before final assembly.
Also, it's a lot easier to cut with precision while the pieces are laying down than trying to do a vertical swipe on a standing wall.
First step is to draw in the door (using razor nicks or marker). The door hinges (in this design) align on the inside of the 1st and 5th units at the base (so the door is 3 units wide). The doors shown here open in the center and you need a middle line for your doors. If you have very sturdy walls, then you might consider a one-side hobbit hole type door.
Using the same center that was used for the roof arch, use a piece of cardboard, tape, or string to hold your razor or marker 1.5 units away from the center. Check that you intersect the hinge lines before drawing the arc.
Use whatever interesting curves you can find around the house to make the windows. And pretty much any round (or other shape) object of appropriate size can be used to make holes for door handles.
Cut everything out and adjust if necessary (it's easier now than later).
One mistake I made was that the door was not quite high enough to clear my daughter's head. So my door had to be cut a little extra after being assembled (and doesn't look as good as it could). Have the user (child or whomever) walk through your wall before final assembly (or use a piece of cardboard to measure them against the door height if you're building this as a surprise.
Another thing I discovered later was that the doors were quite hard to open. Cut a sliver of cardboard to make a gap at the bottom of the doors to keep them from sticking on carpet (if you have carpet). Cut a thin sliver of cardboard from the top arc of the doors to keep them from sticking at the wall (unless you want them to). The spread of the base of the front door from normal has a significant impact on how easy the doors open.

Step 6: Assembly and Reinforcement

I don't have as many pictures as I'd like of the actual assembly because my hands were pretty full. The planks tend to want to slide out (I cut some of my slots too wide). The walls tend to shift and tilt until all the boards are in place. I made it manageable by reinforcing boards as I went to hold the parts together.

(Again, dovetailing would have probably helped quite a bit.)

Have a buddy or wall hold up the structure during assembly. Install the bottom two boards (the bottom slot being the 45 degree cut).
When putting the boards in, pull the folds into the slots from the inside of the structure. Hold them in place and tape into position on the inside and outside of the joints (think L brackets), but don't put tape on the top of the top fold because the next board needs to slide in above the lower board first. You can see how I placed the tape in some of the images.
Install the top center two boards and tape those in position. As soon as two boards are touching side to side, tape the boards together (like was done to secure the box flaps when making the boards).
Install alternating boards on the sides. Keep taping the joints together.
Once I was done, the structure was secure enough to flip over and even pick up at one roof edge without significant distortion.

Step 7: Magic Ingredients

I honestly had intended to flip the front wall so the stickers were facing inside. (Oops. See first picture.) But I'm not about to go back and rip out the flimsy wall just to flip it around. I'll probably give it a light cover of spray paint! I'll probably go classic barn colors (red or blue and something else).

And this brings us to the magic third ingredient. You can add paint, wall paper, windows, flower boxes, interior lighting (I'm thinking LEDs), games, music, solar panels, ham radio... Make it a fun place that you want to visit.

The structure creates a dark cave-like environment so you have full control of the lighting and it probably isn't a bad place for keeping quiet music or game noises from disturbing others. The key structural components of the boards are the folded slotted sections more than the walls, so windows cut into the outer walls should be fine as long as most of the interior board is left intact. If you're concerned about stability, I suggest either punching holes in the interior boards and string them together in a loop (think keychain, but with string). Or if you can find an attractive way to do it - create an outer hoop to hold the boards in (think barrel hoops). I considered putting one wrap of tape around the entire outside of the structure near the back wall, but I didn't want sticky tape inside the barn (even on the floor) and it would look pretty bad (and fortunately I ran out of tape).

Enjoy!!!