You can etch flexible circuit boards using a 3d printer. They can be made of very thin copper clad board material or even conductive fabric.
Standard copper clad circuit board material (FR4) of any thickness, can also be etched using this method.
PLA, Nylon, ABS and most common filaments used for 3d printing do not stick to copper well enough to lay down a pattern that can be etched to create a circuit board. A fairly new elastomeric rubber filament is now available that sticks quite well to copper. It is called Ninjaflex. In fact, it sticks quite well to almost anything including acrylic, blue painters tape, and glass.
A circuit board pattern can be drawn in a free program like 123D Design and then extruded to a thin thickness and saved as an STL file. It can then be printed on top of a thin copper clad board or plated conductive fabric: step 1 pic. It can then be etched in the standard way with a Ferric Chloride solution.
The circuit board shown was printed at standard breadboard and perfboard spacing of traces at .1" centers. This fits standard through hole components and some surface mount ones. Pic 2 shows the finished circuit, a Picaxe microcontroller.
If you want a thin and light circuit, this layout works well. To get more flex, the components would have to be spaced further apart.