Hi Rivière and welcome to permies!
This is a good, relevant question for urban gardeners. I did not know exactly what a "garden tower project" was and had to go search for images. I like the idea a lot, it efficiently allows you to grow some smaller productive plants that you like in a vertical space. What I didn't like so much, however, is the plastic. I don't think the world needs more plastic if it can be avoided for the application, so I was looking for something else, and also something DIY-able as you requested.
I found
this DIY tower, for one. The finished product looks like this:
and the price (they list costs of all materials involved) seems to depend on the cost of cedar or another relatively rot-resistant wood in your area.
The website that the project lives on,
Remove and Replace, also seems to be pretty interesting for permies!
I wonder about some design improvements too, though they would make it harder to build. If the soil trays were not completely flat, but tilted slightly towards one corner, like the earthen roof of a wofati, the water might have a place to go, so the wood would last longer and have less tendency to rot. And a big aircrete mold for one of these might last forever!
But anyway, maybe the project above is good for a start?