Achieving biological softness and flexibility has been a critical challenge for bioelectronic materials used in implantable bionic interfaces and wearable devices. We developed electrically conductive and biodegradable polycaprolactone membranes coated with ultrathin single-walled carbon nanotube via layer-by-layer assembly technique. The resulting membranes have hierarchically structured 3D network of conductive paths around asymmetric nano/micropores which provide softness, stretchability and anisotropic electrical conductivity. The strain sensors embedded on the membrane also demonstrate bending directional sensitivity and piezoresistivity with tunable Gauge Factor in the ranges of 5-13 with stretchability up to 100%. Furthermore, biocompatibility was confirmed by in vitro PC12 cell culture. Therefore, this implantable multifunctional membrane has considerable potential as a conductive biomaterial for bioengineered devices such as mechano-sensitive artificial interface.