A magnetic field is not a thing unto itself, but a region of space subtended by a force. You could just as well ask: what is a force made of? well, it's not a thing either, so it's not made of anything. A force is a state of matter. Matter is the thing. Matter has mass and charge and spin - these are not things either but they are the cause of force.
So a magnetic field is simply the manifestation, in spatial terms, of the attributes - mass, charge, spin - of the things, which are particles or atoms or bodies that possess the qualities of mass, charge, spin (etc.)
Fields are densities, for example Webers per meter or Volts per meter. Notice, you would not think to ask this of forty-weight motor oil. 40W is a density. So imagine asking: what is a 40W made of? See, it doesn't make sense.
The notion of fields has been popularized by Sci Fi (and maybe Sci Fo!) so that we have all kinds of artistic renderings of it in our mind. These renderings tend to slip into the holes where we haven't covered some uncertainty with a sanity check, something to keep us grounded in reality before we lose touch with basic principles that hold everything together.
Having said all of this, I am also interested in how fields arise, how they propagate and what is going on with virtual particle exchange, and the underlying mechanisms, so if that was where you were headed, I'll kick back and listen.