Six years ago, I had to make a bed base so our mattress wasn't sitting on the floor. I had lots of plastic bottles that needed recycling, and an idea.

Step 1: Tools and Materials

The reason this bed is so primitive is that I had just moved countries and had no power tools with which to make one, except my first purchase, a cordless drill. That's all you'll need, because you can get the plywood cut to size when you buy it (home centers and lumberyards all have panel saws for exactly this purpose).
5/8" (or thicker) plywood cut to the size of your mattress. This will probably involve two sheets. Buy the cheap stuff or recycle some, no one is going to see it. Measure your mattress carefully first! Keep the offcuts for future projects.
Lots of plastic bottles. They don't have to be the same brand, but do need to be the same height and diameter, made of PET, and have plastic lids. I used 710 ml bottles of Canada Dry - the previous owner of my house was clearly a fan. You could, of course, make a higher bed by using bigger bottles.
Screws - at least 77, shorter than the thickness of your plywood. I found a box of 1/2" self-tapping square drive screws, so used them. You'll also need 2-3 longer screws.

Step 2: Join the Plywood

If your bed is bigger than a single, you'll be using two pieces of plywood. Butt these together, and join. I had some mending plates lying around, so used those, but pieces of wood or nailing plates would work fine too. Make sure you leave the good face of the plywood facing up.

Step 3: Mark Out the Leg Positions

I ended up with an eleven-legged bed, because I had a mountain of soda bottles to get rid of. 9 legs is probably plenty. Each leg is made up of 7 bottles. Make yourself a template by balancing 7 lidless bottles upside down on a piece of paper, and trace around each neck. Put one in each corner, middle of edge, and center of bed, as shown. The extra 2 at the head of the bed as shown here are probably unnecessary and reduce the amount of storage under the bed for suitcases etc.

Step 4: Screw the Lids to the Plywood

Remove the plastic liners from the lids. Drill a hole in each lid, then screw in place. Repeat 76 times.

Step 5: Add the Legs

Screw the bottles into the caps, starting with the central one for each leg. Hold them together by duct-taping the six outer ones together.

Step 6: Ready to Go

Your bed base ought to look something like the picture below. It is very light, and has no bracing to speak of, so you'll have to screw it to the wall. Otherwise it will wobble like crazy and probably end up failing. I drove some deck screws at 45 degrees through the plywood and into some wall studs.