What's an electron made of? What's a neutrino made of? If they are really fundamental particles, then the whole point is that they are not made of anything more fundamental. If however string theory is a theory of nature, then electrons are made of string. Now we are back to step 1. If the string is fundamental, then it's not made of anything else, that's the whole point of being fundamental. Of course if M-theory is right, then the string is made of membrane. And we are back to step 1. What's the fundamental membrane made of? If it's fundamental, then it's not made of anything else. On the other hand, if AdS4/CFT3 is right, then the membrane is sort of made of string. At which point you may feel confused.

More seriously, it is important to realize that when we get to fundamental physics, then everday concepts of existence are far behind us and all there remains is some mathematics and a way to deduce scattering cross sections from it. Perturbative string theory is a piece of mathematics that spits of scattering cross sections. It does so by computing correlators of some given 2d superconformal field theory and integrating them over the moduli spaces of super Riemann surfaces, for all genera. That's one big formula, called the string perturbation series, but whatever it actually does, in the end it spits out elements of an S-matrix. And there is a rule for what it means to compare these to scattering experiments. Given that, before you even ask "what is the string made of" you should ask "where do you see a string in the first place"? Ultimately, speaking of "1-dimensional objects propagating through spacetime" is a story that helps think about what that big perturbation series formula actually does. Of course it's a very good story, and so it's used all the time, and as with all stories, after a while one gets a feel for it and it seems obvious and real.

But asking "what's the string made of" is much like asking for the reality of virtual particles. Ultimately there is just a formula, now called the Feynman perturbation series, and it does something and spits out elements of an S-matrix which are being compared to experiment. Virtual particles are just a story that is told which helps think about what this big formula does. And it's a very good story, and so it's used all the time, and as with all stories, after a while one gets a feel for it and it seems obvious and real. But no matter how real it seems, the virtual particles in QFT perturbation theory are a figment of our imagination that helps navigate the realm of fundamental physics which is far, far away from what our minds evolved to being able to visualize. In the end there is just some mathematics which spits out observable numbers. And then people tell a story around it.