A table saw is the right tool for this, and you don’t need a dado blade, although its pretty much what they are for. A standard table saw blade will cut at right at 1/8th inch. Proper dado blades are for rich people. You might also use a router in a router table, but router tables are pretty single use tools and if you arent’ getting a ton of use, its not at all worth it. I would make this cut on a table saw, depending on the size of the piece.

And if you need to go larger than 1/8th… You set your fence to the far or exterior edge of the dado you want to make. Then you set your blade height to whatever you want. Then advancing about 1/16th of an inch at a time, you just move your fence towards the direction of the dado (which would towards the blade), 1/16th or so at a time, doing a series of rip cuts. As long as you’ve got the blade height nice and locked in and you have patience, should get a consistent depth, which you can clean up afterwards with a chisel. Doing it this way you can make a dado of any size, and you have very fine control over the dados. This is pretty much exactly what I’ve done in the past to mount 3/16th inch plywood into box bottoms. I took the entire board I was going to be cutting down into box walls and made one dado that went all the way down at 3/16ths. Like 2-4 adjustments to the fence and I had a perfect 3/6th dado, and when I cross cut the board to the right sizes for the box, my dado’s were all in perfect alignment.

Alternatively, you could use either a circ saw or a router. It really depends on the size of your piece and how many of these dados you have to make. But you can mount another piece of wood (spare 2x4 or plywood) using clamps, and run either a router or a circ saw. If the piece you are trying to do this on is like, very large, you may have to do it this way. In this case, I would prefer the router. Circ saws just don’t cut that straight. A circ saw would be my last choice.

So order of preference for me to make this kind of cut:

table saw > router table > router with clamps and another piece of wood as guide > circ saw with clamps and another piece of wood as guide.