Yakov and Moshet Pedagog’s living room, with its soft armchairs, air conditioning and panoramic view of central Israel’s hills, seems the perfect place for a couple in their 80s to exhale after long careers in education.

But Yakov, 85, and Moshet, 82, have no such plans.

“We’re not planning to relax,” said the American-born Yakov, who immigrated to Israel with the aliyah organization...