This lead me to the following - what if you make a compost pile the size (mass) of Sun? Meaning, it won't be made of hydrogen, but rather some carbon-based molecules. I'm not sure about other atoms in these molecules, but I think carbon is stable enough not to initiate nuclear reactions. So probably fusion won't start. What then?

Gravity would compress it. Depending on the exact composition fusion could start, ie it would become a star. According to Wikipedia carbon fusion requires a mass of 8 suns to start but there's a bunch of hydrogen in compost so maybe that would fuse.

Carbon is capable of fusion, in a sufficiently compressed stellar mass. Everything up to Iron-56 can theoretically sustain fusion.

Maybe it could work via the CNO cycle: https://en.wikipedia.org/wiki/CNO_cycle ?

Does the volume of the sun include its Corona in that last statement?

No, iirc that statement is for the part of the sun where fusion actually happens. Per volume, not a lot fusions happen in the plasma on the sun. The crux is that the volume of the compost heap grows order cubed to the radius, while the area through which the heat escapes grows quadratic. And the sun is like, a _really_ big compost heap.

In fusion on earth, we want to be considerably more efficient than the fusion process in the sun, as we don't have as much space to work with. ITER is already a pretty big machine.

That’s just the core. Go out to the photosphere, let alone the corona, and it’s about about a thousand times less, 0.27 W/m^3: https://www.wolframalpha.com/input/?i=%28solar+luminosity+%2...