Hampton Chambers, the would-be patriarch played by David Oyelowo in the Apple TV+ series “Government Cheese,” bears some resemblance to a classic sitcom dad. He has moved on up, finding a home in a tidy San Fernando Valley suburb for his wife and two sons and striving to keep them there. He is obsessed with taking family photos. He cajoles his rebellious younger son into a weekend fishing trip at a nearby lake, with predictably comic results.

But what distinguishes Hampton, and “Government Cheese,” are the ways in which he departs from the stereotype. The fishing trip is a cover for a nighttime burglary. The photos are exculpatory evidence. He owes a debt to a local criminal clan of seven thuggish French Canadian brothers. His George Jefferson cockiness is cracked by fissures of guilt, fear and regret; he pleads with Yahweh for forgiveness.

That might make Hampton sound like a Walter White (“Breaking Bad”) or a Marty Byrde (“Ozark”), losing his way under pressure. But “Government Cheese,” which premiered Tuesday with three of its 10 episodes, is indeed a comedy, if a barbed and mysterious one; it straddles a border between the pioneering Black sitcoms of the 1970s (it’s set in 1969) and the fable-like dramedies of the streaming era, particularly “Lodge 49,” a show it strongly evokes. (There is also some “Fargo” in it, at the darker end.)

“Absurdism” and “surrealism” are the words Apple TV+ has applied most liberally in the show’s publicity materials. American comedy very rarely commits to these qualities, though. What “Government Cheese” really offers is something softer and more common: a mildly sardonic, artfully presented magical realism.

Hampton is in prison for petty fraud when the show starts, about to be paroled. He comes home to find that his family has been hijacked by the ‘60s. One son, Einstein (Evan Ellison), is a soft-spoken prodigy who sees his future in pole vaulting; the other, Harrison (Jahi Di’Allo Winston), is a budding radical who wears a Billy Jack hat and identifies with the Chumash people. Hampton’s wife, Astoria (Simone Missick), has a job and a man on the side, and sees her husband’s return as a threat to her tentative freedoms.

Hampton’s hopes of proving himself and keeping his family together are pinned to a gizmo he invented in the prison machine shop, a self-sharpening drill. (It isn’t clear whether it actually works.) In his way are the comically violent Prévost brothers, as well as the everyday difficulties of being out of place, as an ex-con and a Black man, in both suburban Los Angeles and the aerospace industry. That’s why he reluctantly puts the drill to use as a safecracking tool, in league with his old friend Bootsy (a jovial Bokeem Woodbine).

It is the show’s re-creation of this time and place in which Hampton feels himself falling behind that holds viewers’ interest. Paul Hunter, a prolific director of music videos who created the show with Aeysha Carr, grew up in the Valley, and the Southern California ambience is heady: spirituality as a self-improvement program; the rough-and-tumble aerospace plants; nearby orange groves and casual water theft. Looming in the background of many shots are the Chatsworth rock formations, a silent reminder that in the year the show is set, the Manson family’s base of operations was just up the road.

Hunter and Jim Gavin, the creator of the 2018-19 “Lodge 49,” both used checkered family histories in Southern California as inspiration for distinctive, highly personal series. Their shows share a fable-like approach, perhaps a way of rendering the burnishing glow of memory; a fanciful episode of the new show set in an Elks Lodge even feels like an homage. “Government Cheese” doesn’t have the consistency of tone, the melancholy eccentricity that “Lodge 49” had; its impulse is more satirical, less curatorial. But characters vanish and animals assume mythic dimensions. Hampton, in search of atonement, finds himself in a Jonah-like situation that is definitely not naturalistic.

(Surplus government cheese does not make an appearance in any overt way. The title appears to be emblematic of the period, and perhaps of the idea of playing the system to get by.)

The palpable intelligence that Oyelowo brings to his roles is a good fit for Hampton, whose most salient attribute is his scrambling ingenuity. (That, and the self-centeredness he claims to be trying to overcome.) If Hampton ends up feeling a little blank, it may be because Oyelowo has more technique than pure magnetism, but it also may be because the show doesn’t seem sure yet about its attitude toward its protagonist and the possibility of his redemption in seasons to come.

There is plenty of charisma on offer, though, from Ellison, Winston and particularly the wryly self-assured Missick (of “All Rise”) as Hampton’s loving but impatient family. We may wonder where he’s going, but they’re not waiting around for him to figure it out.