BETHESDA, Md.—Olympic swimming champion Katie Ledecky has broken records in pools around the globe. But in this affluent Washington, D.C., suburb, not even gold medals are enough to win quick entry into the Little Falls Swimming Club.

Using an arcane set of rules, the club each spring welcomes a few newcomers into its membership, which is capped at 377 families. Competition is so intense that residents have been known to choose homes based on proximity to the facility.

The annual selection process uses a numerical formula that gives would-be members points based on 10 concentric rings of increasing distance between an applicant’s residence and the exact center of the pool. The closer they are to the club, the better.

Other considerations include time notched on the waiting list and whether a family owns or rents. Most applicants spend six to eight years in pool purgatory.

Ms. Ledecky’s family first applied about 15 years ago, before the swimming icon joined a team. Like all other hopefuls in the area, they had to wait.