I have a GIS dataset currently projected in Web Mercator.
Is there an option for a more accurate segmentation length projection when I need accurate lengths in Southeast Korea?
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If I understand correctly you want to re-project your data? Check https://epsg.io/ for appropriate projections for your study area. For example, EPSG:5179 might work.
Depending on which version you're using, see if Calculate Geometry Attributes tool is available. It can calculate the geodesic length of features.