Direct laser writing (DLW) via two-photon polymerization (TPP) has been established as a powerful technique for fabrication and integration of nanoscale components, as it enables the production of three dimensional (3D) micro/nano objects. This technique has indeed led to numerous applications, including micro- and nanoelectromechanical systems (MEMS/NEMS), metamaterials, mechanical metamaterials, and photonic crystals. However, as the feature sizes decrease, an urgent demand has emerged to uncover the mechanics of nanosized polymer materials. Here, we fabricate coil spring shaped polymer nanowires using DLW via two-photon polymerization. We find that even the nanocoil springs follow a linear-response against applied forces, following Hooke's law, as revealed by compression tests using an atomic force microscope. Further, the elasticity of the polymer material is found to become significantly greater as the wire radius is decreased from 550 to 350 nm. Polarized Raman spectroscopy measurements show that polymer chains are aligned in nanowires along the axis, which may be responsible for the size dependence. Our findings provide insight into the nanomechanics of polymer materials fabricated by DLW, which leads to further applications based on nanosized polymer materials.