Some extremely low-dimensional yet crucial geometric eigen-lengths often determine the success of some geometric tasks. For example, the height of an object is important to measure to check if it can fit between the shelves of a cabinet, while the width of a couch is crucial when trying to move it through a doorway. Humans have materialized such crucial geometric eigen-lengths in common sense since they are very useful in serving as succinct yet effective, highly interpretable, and universal object representations. However, it remains obscure and underexplored if learning systems can be equipped with similar capabilities of automatically discovering such key geometric quantities from doing tasks. In this work, we therefore for the first time formulate and propose a novel learning problem on this question and set up a benchmark suite including tasks, data, and evaluation metrics for studying the problem. We focus on a family of common fitting tasks as the testbed for the proposed learning problem. We explore potential solutions and demonstrate the feasibility of learning eigen-lengths from simply observing successful and failed fitting trials. We also attempt geometric grounding for more accurate eigen-length measurement and study the reusability of the learned geometric eigen-lengths across multiple tasks. Our work marks the first exploratory step toward learning crucial geometric eigen-lengths and we hope it can inspire future research in tackling this important yet underexplored problem.