Line tension, i.e., the force on a three-phase contact line, has been a subject of extensive research due to its impact on technological applications including nanolithography and nanofluidics. However, there is no consensus on the sign and magnitude of the line tension,-mainly because it only affects the shape of small droplets, below the length scale dictated by the ratio of line tension to surface tension sigma/tau. This ratio is related to the size of constitutive molecules in the system, which translates to a nanometer for conventional fluids. Here, we show that this ratio is orders of magnitude larger in lyotropic liquid crystal systems comprising micrometer-long colloidal particles. Such systems are known to form spindle-shaped elongated liquid crystal droplets in coexistence with the isotropic phase, with the droplets flattening when in contact with flat solid surfaces. We propose a method to characterize the line tension by fitting measured droplet shape to a macroscopic theoretical model that incorporates interfacial forces and elastic deformation of the nematic phase. By applying this method to hundreds of aroplets of carbon nanotubes dissolved in chlorosulfonic acid, we find that sigma/tau similar to-0.84 +/- 0.06 mu m. This ratio is 2 orders of magnitude larger than what has been reported for conventional fluids, in agreement with theoretical scaling arguments.