This is how I made a low-profile bed frame cheaply from wood slats.
We want our mattress low to the ground so if our daughter ever falls out of bed, she doesn't have far to go. We learned the trick of putting the family bed on the ground from Christy's parents, and wanted to give it a try. However, the floor in our bedroom is uninsulated, and putting a mattress directly against a cold floor is a great way to grow mold. This frame raises the mattress 1.5 inches; just enough to ensure airflow.
Step 1: Build the Framing with Furring Strips
My local Home Depot had 1x3 8 ft pine furring strips (0.75 inches x 2.5 inches actual) for $1.22 each. These things aren't particularly straight or high-quality, but they're perfect for my purpose. I bought 23 of them and a box of 1 1/4 inch nails for a total cost just under $30.
I measured my mattress, then cut all the furring strips to the proper length with a chop saw.
Step 2: Layout and Nail the Frame Together
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Lay all the pieces out, and nail it together. I used one of the furring strips as a spacer as I attached the slats to the frame.
Note that if you only put one nail at each intersection, the whole thing can rack like a giant pair of scissors or a folding trellis.
Step 3: Attach Felt Feet
I attached felt feet to the frame because I planned to place it directly on a hardwood floor.
Step 4: Mattress on the Frame
Here's the mattress on the frame. Airflow underneath the bed is only occasionally blocked by bedding, and never on all four sides at the same time.
Here's Corvidae climbing off the side of the bed. When she manages to go head-first instead of her usual feet-first or hands-first, the resulting bump only requires some brief comforting, not x-rays. We often put blankets and pillows around the edge of the bed for an even softer landing. She can also crawl into the bed by herself and fall asleep.