It’s easy to approach a task, see that you don’t have the proper tool for the job, and write it off as an impossible task. It takes creativity and some persistence, but you can actually build custom cabinets without a cabinet saw.
I was tasked with making a custom cabinet 16 1/2″ deep and 28″ wide at standard base cabinet height. To achieve maximum strength, I needed to cut two dados in two panels: one for the backing, and one for the bottom shelf. I set about accomplishing the task using factory edges on sacrificial plywood pin nailed to the work to create a straight guide for my circular saw. To cut the first shoulder of the dado, measure the distance from your blade to edge of fence over and tack your guide piece in place. Set your depth to 1/4″ or so. Cut your first groove. For the second, use a scrap of the shelf material to sit between your circular saw fence and your guide to manage a perfect spacing for your dado. Once cut, pry the waste out with a wide timber framing chisel.
For your backing, space the dado in enough to allow for a cross piece from the panel material. In my case, 3/4″. With dado cut, pin nail together, patch any imperfections, and the cabinet is finished. Budget 3-4 hours labor and 1 sheet of ply for a single cabinet at $35-45 per sheet.