At a certain level of enduring fame, it becomes difficult for an actor to be taken seriously in a new artistic sphere. By the time someone is both a beloved global star and an avatar of the American good guy — yes, I do mean Tom Hanks — this rule has long since been in force.

That didn’t stop Hanks from publishing “Uncommon Type,” a book of his short stories, in 2017, and it hasn’t stopped him from his current project: starring in “This World of Tomorrow,” a new play that he and James Glossman adapted from some of those tales.

The prospect of the production, at the Shed, might understandably have been greeted with skepticism — even though the director, Kenny Leon, and Hanks’s co-headliner, Kelli O’Hara, are Tony Award-winning theater veterans, and the supporting cast looked strong. Was this going to be a case of good actors trapped in a bad play written by a boldface name?

It’s more of a comfort-food experience, actually: a time-travel romance that often feels like a period musical minus the songs. Hanks and Glossman aim to entertain while underlining some fundamental American values and the importance of building a better future. Aided by Leon’s smooth direction, they succeed at that.

At ease onstage, Hanks (in a distracting wig by J. Jared Janas) stays very much within his bailiwick here. He plays a wealthy tech entrepreneur from 2089 with an unlikely throwback name, Bert Allenberry, and a history nerd’s relish for the past. Time travel being, apparently, primitive, he opts repeatedly for one of the few date-time combinations available: June 8, 1939, in New York, where he always arrives in Room 1114 of the Hotel Lincoln in Manhattan, and where his attempts to follow the local customs are endearingly game.

Handing a dollar to a room service waiter, he hazards awkwardly: “And, ‘keep the change’? Is that what I am supposed to say now?”

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