This study leverages the ancient craft of weaving to prepare membranes that can effectively treat oil/water mixtures, specifically challenging nanoemulsions. Drawing inspiration from the core-shell architecture of spider silk, we have engineered fibers, the fundamental building blocks for weaving membranes, that feature a mechanically robust core for tight weaving, coupled with a CO
2
-responsive shell that allows for on-demand wettability adjustments. Tightly weaving these fibers produces membranes with ideal pores, achieving over 99.6% separation efficiency for nanoemulsions with droplets as small as 20 nm. They offer high flux rates, on-demand self-cleaning, and can switch between sieving oil and water nanodroplets through simple CO
2
/N
2
stimulation. Moreover, weaving can produce sufficiently large membranes (4800 cm
2
) to assemble a module that exhibits long-term stability and performance, surpassing state-of-the-art technologies for nanoemulsion separations, thus making industrial application a practical reality.