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.