Back in the 1950s, when John Rubinstein was 8 or 9 years old, he was granted a brief in-person meeting with the president, Dwight D. Eisenhower. Mr. Rubinstein’s father, the legendary pianist Arthur, was in town for a concert, and the president’s chief of staff, Sherman Adams, had arranged for a tour of the White House, culminating in a short chat with the big man himself.

“He and my father had known each other as acquaintances,” Mr. Rubinstein, now in his 70s, recalls of Eisenhower, who excused himself from another conversation to greet the renowned musician and his wife before turning to their children. “It was my sister and me; I was wearing an ‘I Like Ike’ button. And he just said to me, ‘Hi, little fella — nice to meet you.’”

About seven decades later, Mr. Rubinstein, a stage and screen veteran whose notable Broadway credits include the lead role in Bob Fosse’s original production of “Pippin” and a Tony Award-winning turn in “Children of a Lesser God,” is portraying our 34th president in “Eisenhower: This Piece of Ground,” by Richard Hellesen, who drew on Eisenhower’s own writing and writing about him. The production recently began a return engagement off-Broadway, due to run through October 27.

The two-act, one-man play, which had its premiere at Los Angeles last fall, is set in 1962, nearly two years into Eisenhower’s retirement following a political and military career that had also included leading the Allied forces to victory in Western Europe during World War II. While preparing to write a book detailing his two-term presidency, he is flummoxed by an article in the New York Times Magazine — one that Mr. Rubinstein remembers reading while in high school — in which historians rank him at just no. 22 among American presidents. (That perception has changed through the years; as the play eventually documents, a similar poll from 2022 had Eisenhower at no. 5.)

Much of the monologue that follows finds Eisenhower speaking into a tape recorder, revisiting past accomplishments as well as a few regrets. He refers proudly to achievements such as establishing an interstate highway system, integrating the military, and signing the Civil Rights Act of 1957, and trying to scale down the nuclear arms race. He defends himself against various accusations — that he could have done more to bring down Senator Joseph McCarthy, whom he loathed, or that the U-2 incident that sabotaged his budding negotiations with the Soviet premier, Nikita Khrushchev, was his fault.

Mr. Hellesen also digs into more personal aspects of Eisenhower’s life, from his affection and concern for the soldiers who served under him to his marriage to Mamie Doud, as well as the early loss of the first of their two sons to scarlet fever, when the boy was just 3.

Mr. Rubinstein admits he was hardly an expert on this history when director Petter Ellenstein first approached him about “Eisenhower.” His youthful encounter with the president notwithstanding, Eisenhower’s successor had done more to fire his imagination: “Kennedy was the guy. Eisenhower was the old guy; he was bald and he played golf.”

Mr. Rubinstein also had doubts about the show’s appeal: “A play about Eisenhower, with one actor? That’s a hard sell, unless maybe it’s Brad Pitt.” Yet he had been interested in doing a one-man show, and after agreeing to a reading, he began listening to tapes of Eisenhower delivering speeches. “I realized he sounded exactly like a person I knew pretty well, who’s now deceased. He was a high school principal in Indiana, so he sounded authoritative; he was used to being listened to.”

With that voice in his ear for reference — “It wasn’t a direct imitation,” Mr. Rubinstein stresses — the actor proceeded to read for Messrs. Ellenstein and Hellesen, “and by the end, all three of us were like, ‘We’ve got to do this play.’ Because when you read it aloud, there’s so much in it, and it’s so moving. It’s very informative, but it doesn’t come off like a history lecture.”

Mr. Rubinstein found himself particularly drawn to Eisenhower’s concerns about the threats of isolationism and extremism, and his belief in “a middle way,” as the politician puts it. “If you don’t know the difference between government domination and government doing a job that needs to be done, I can’t help you,” he muses at one point in the play.

“That’s what I found startling — that he was dealing with versions of things we’re still dealing with,” Mr. Rubinstein says. The actor says he would love to bring “Eisenhower” to national audiences were the opportunity to tour arise. “But that’s not up to me — someone has to book it,” he points out. “Eisenhower” will be returning to California, to Ventura’s Rubicon Theatre — “I can drive there from my home,” he notes — in February.

Bringing one of the 20th century’s most monumental figures back to life in a solo performance, six or seven times a week, has been challenging, Mr. Rubinstein admits. “I go home and it’s hard to sleep … but I love this play; I love what it says. And like anything else, it gets a little bit easier the more you do it. And every time I do it, I learn something new.”