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Assuming you have the basic skills with the tools you'll be using, there are a few ways to do this. First, you could do this with a handsaw. That would be your monthly workout, but it is a method, and it would work.
You could use a jigsaw, but you will have edges to work on. If you have access to a circular saw, that would work with certain jigs. Cutting the narrow sizes may get a bit difficult. Or, you could mount the circular saw to the underside of a substrate of sorts, like melamine, and use a straight piece of plywood for a clampdown fence.
So, what you've created is an archaic form of a table saw. Not as easy to use as a real one, but it will work. You might consider renting an equipped table saw for a day or two. Or, contact some cabinet shops in the area and see if they will cut your pieces.
If you doubt your abilities, seriously reconsider the project. If this will be your one and only woodworking project for the rest of your life, you have here a few suggestions. If you plan on other projects, getting a good tablesaw would be a wise decision.