They don’t seem to be a complicated design. However, I don’t have any appropriate tools really, just what I’ve collected through the years as a renter. So, does anybody have an idea what the minimal amount of tools I’d need would be to make something like the photos below? Fine saw? Corner clamp? Mitre guide?
To do it like this you would need a mitre saw, very well set up, to do the mitres.
A router to do the dado / groove for the drawer base
Some clamps and glue to stick it together. You can buy corner clamp guides to make sure it's all square, or you can make a template.
and some oak veneered MDF (it's usually veneered on one face and one edge), and some plain MDF for the bases, all of which you can order cut to size.
But as others have said, a mitre like that is not a strong construction, so there may be something hidden we cannot see, and getting that level of finish takes time, skill and practice.
Have you looked on Ebay, Facebook Marketplace or Gumtree for any of these for sale? It's probably a cheaper and easier way to get more drawers. I've pimped loads of old Ikea things by simply buying more second-hand ones and using what I need from them.