My son has a lot of toys. For that purpose, I would like (if durable suitable drawers can't be bought) consider making drawers or boxes on rollers in a frame or something like. I live in Japan and we have dining room table and almost no tools beyond a small hammer and small screwdriver. There is decent hardware store in our area. The space I would like build into is 70 cm wide by 100 cm deep and 150 cm- I am totally on the number of drawers to build. There certainly lots of drawer building video which focus on small aesthetic pleasing desk drawers. If I really don't care about aesthetics (I don't care how it looks), is the project practical if I have never really built anything? What about I bought plastic boxes and then mounted them on a frame?
Rather than buying a tablesaw, you could buy a Japanese hand saw and a chisel. With these you can certainly build a box and with practice, a cabinet and a set of drawers.
Japanese saws cut on the pull stroke, so they can be made thinner than European saws. This makes them ideal for dovetailing.
For the box you will only need to learn how to do one joint - either dovetail or box joint. Or even a butt joint and just screw the walls together.
If you decide on a dovetail, remember that with hand woodworking you can make the tail as large as you like. There is no need to do twenty 1cm dovetails on each side - although this would be good practice!
Do not be shocked by the chisel prices. That is for a full set. Initially you only need one as wide as the joint you are making.
In the linked video, don't worry about all the planing. If you can get some reasonably straight and flat wood, for a toy box you should be fine. He uses a really easy joint that only needs a saw.
To make the bottom of the box, just nail some plywood across.
If you find you enjoy woodworking, The Anarchist's Tool Chest by Chris Schwarz will give you a box to put all your tools in. Chris is an excellent place to go for learning about woodworking.
The one tool that may do everything you need to build drawers is a tablesaw. The bigger the better, width cutting capacity wise, that is. With repeated passes you can cut dadoes or rabbets or even finger joints for drawer sides. Rabbets for the bottom to reside in, besides sizing the material for assembly. If you learn what you need to apply the tool to what you want to make, it will be quite feasible. First projects aren't usually great looking to begin with although they could, the esthetics improve over time after gaining experience on doing more projects. Who knows, you may like it and want to do more over time.....
It is certainly possible to build good drawers, as @aldante describes, but one can also (at least in North America, and I assume in Japan) order custom sized drawers online that arrive as easy to assemble parts.
Cost-wise, they might be appealing when compared to buying and storing basic tools. (And of course, assembly time is minimal compared to building from scratch.)