BUENOS AIRES—It was a familiar scene for Argentines. As a financial crisis engulfing the country entered its fourth week in early September, residents were lining up at banks to withdraw dollars from their accounts, hoping to keep their savings safe.

They had yanked close to $6 billion when President Mauricio Macri’s government took emergency measures. After implementing currency controls, it tapped the central bank’s dollar deposits at the U.S. Federal Reserve, airlifting some $4 billion in cash packed in compressed, heat-sealed...