Totally possible. There's a few parts on that particular cabinet that were probably cut with a table saw (the little blocks holding up the shelves for example) but there are ways to get around that.
Miter saws are great but limited in what they can do. One, they only crosscut (they can't rip the length of a board) and two they're not the most accurate tool as the blade likes to deflect (this can be mostly solved by spending time to square up the saw properly, investing in a good blade, and using good technique). That's why the table saw is kinda the cornerstone tool for most woodworking (you can build jigs to allow a table saw to do damn near anything) but like I said - there's ways around it. For example, a circular saw and an edge guide can make damn good rip cuts if you take your time with setup.
That cabinet is really just a matter of building a box. There are a bajillion different ways to build a box depending on how strong you want it to be, what you want it to look like, whether or not you care if screws are visible, etc. For the most part there's no right or wrong answers, just a matter of how complicated you want to make it. About the only thing you really want to avoid is gluing the end grain of a piece of wood without some other method to reinforce the joint (because the glue seeps into the fibers of the wood). The corners of the box are really where this is an issue. One easy/fast way to deal with that is pocket holes (only requires a drill and a pocket hole jig) but you'd end up with visible screws on the inside. Another good option could be dowels (again, a drill and a simple jig are all you'd need). Or you can get into fancy joinery like the box joints I poorly implemented in my medicine cabinet but most of that is easiest with a table saw.
As for materials, you could 100% build that out of materials from the big box stores. You could build the shell from plywood, or from boards of whatever wood you like, paint it stain it, etc. Or you could go to a specialty hardwoods store and build it out of high end hardwood veneered plywood or even solid hardwoods, domestic or exotic. Again, it's all a matter of what you want it to look like and how much you want to spend.
This video I think is pretty helpful:
https://youtu.be/p1aQQQDXrc8?si=-pWi3rHE2xPVcW0t
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