MOSCOW — A stream of elegant visitors stopped in their tracks on Thursday as they toured Moscow’s new Jewish Museum and Tolerance Center, a sprawling, state-of-the-art complex underwritten by oligarchs close to President Vladimir V. Putin. They had never seen a shtetl like this one.

Touch the screen in one exhibit in this vast building and a visitor can appear in a mirror dressed in the garb of a 19th-century blacksmith, or a trader or a “representative of the intelligentsia.” Tap a Torah in a virtual synagogue, and a cantor’s voice rings in the air. In a virtual Odessa, one can sit down in an interactive cafe to chat with long-dead writers.

Mr. Putin has extended his personal support to the lavish project, donating a month’s salary for its construction, which cost around $50 million. In part because of its scale — organizers say it is the largest Jewish history museum in the world — the project is meant to convey a powerful message to Jews whose ancestors fled or emigrated: Russia wants you back.

President Shimon Peres of Israel, who attended the opening, said it affected him deeply....