Current advances in manufacturing nanotubular and nanofibrous materials with high surface- to - volume ratios call for the development of adequate characterization methods and predictive estimates of their absorption capacity. Extremely high flexibility of these materials poses a challenge: their pore structure easily changes upon contact with the fluid in question. One more complication is that the absorption process is sufficiently fast: one hundred micron droplets disappear in milliseconds. This paper sets a physical basis for analyses of absorption processes in nanotubular and nanofibrous materials. As an example, we study the absorption of droplets by yarns made of nanofibers and nanofibrous webs. Through our experiments, we show that absorption can induce different types of nanoyarn deformations: visible deformations of the yarn profile and deformations of the yarn diameter/length caused by the capillary pressure. Using our experimental data and theory, we estimate elastic and transport characteristics of the nanofibrous materials. The reported experiments and proposed theory open a new area of research on absorption-induced deformations of nanotubular and nanofibrous materials and show their potential applications as sensors to probe minute amount of absorbable liquids.