Postgres, Connections to all databases refused

I was recently asked to look at an interesting problem in which nothing can connect to Postgres — to any databases managed by Postgres. All connecting clients would get this rejection error message:

[postgres@svsdssib09 ~]$ psql -d postgres -c '\l'
psql: FATAL:  database "postgres" is not currently accepting connections

Connections to any database, including the template0 and template1, were rejected.

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