We present the fabrication and electrical characterization of a flexible hybrid composite structure using aligned multiwall carbon nanotube arrays in a poly(dimethylsiloxane) (PDMS) matrix. Using lithographically patterned nanotube arrays, one can make these structures at any length scale from submicrometer levels to bulk quantities. The PDMS matrix undergoes excellent conformal filling within the dense nanotube network, giving rise to extremely flexible conducting structures with unique electromechanical properties. We demonstrate its robustness against high stress conditions, under which the composite is found to retain its conducting nature. We also demonstrate that these structures can be utilized directly as flexible field-emission devices. Our devices show some of the best field-enhancement factors and turn-on electric fields reported so far.