"I've met Idris Elba," says June Squibb, an edge of giddiness in her voice. "I've met Lupita [Nyong'o] and now we're friends. She's so sweet, a darling girl. I went up to her at a party and said: 'I'm June Squibb and we're in this together so I thought I should introduce myself.' Julia Roberts came and introduced herself to me – she's up for supporting this time around for Osage County." The day before we meet, Squibb went to the nominees' lunch. "All 270 of us got our picture taken together. I was right by Steve McQueen!"

Squibb, 84, but feeling like a first-timer, is nominated for best supporting actress this year for her diamond-sharp role in Alexander Payne's melancholy road movie Nebraska. She's Kate, the abrasive wife of Bruce Dern's forgetful wanderer Woody Grant, who is driving cross country in search of an elusive million-dollar prize. Kate is what they used to call a doughty plains-woman, a squat, waddling fireplug of a grandma too old to care what comes out of her mouth or whom it offends. In a characteristic moment she stands with Woody over his dead sister's grave and carps in her piercing, querulous tone: "Oh, she was such a whore!" She's a chaotic, deliciously annoying life-force in the movie, forever chivvying along her wayward husband and her two glum sons (Will Forte and Bob Odenkirk). Elsewhere, recently, Squibb – perhaps proving there are better opportunities, to say nothing of a thinner field, for female actors over 80 than for those over 40 – has played Hannah's dying grandma in Girls and, in the HBO adaptation of Getting On, a surprisingly knockabout physical performance as a nasty, racist old redneck off her meds, given to puking into nurses' cleavages and screaming: "I'm from Bakersfield and I want my fucking dinner!"

That is decidedly not the lady I meet today. Squibb glides into the room with a svelte grace no doubt engrained by years spent singing and dancing in 1950s musical theatre. She is dressed rather regally in black, with elegant no-frame glasses, her white hair fluffed into a fuller bob than the tragic skullcap hairdo she gave Kate. And the voice is much softer and sweeter, uncannily like Ellen Burstyn's, with a tinkling laugh that rises to a little gurgle when talk turns risque or gossipy.

Nebraska is the second time that Squibb has been called on by Alexander Payne to play the wife of an ageing screen legend. Twelve years ago, it was her character's death early in Payne's About Schmidt that propelled Jack Nicholson on his improbable odyssey in that movie. Years later the call came again: "Alex called me up and said I gave you one icon in Jack Nicholson and now I'm giving you another in Bruce Dern," she laughs. "And somewhere in between I was in Welcome to Mooseport, which turned out to be Gene Hackman's last movie before he retired."

I ask her about working with Dern, an actors' actor for half a century, and also a respected teacher of acting. His performance is a miracle of concealment and non-revelation that seems to force the other actors all to reach toward him. "I think that actually Bruce and I work very much alike," she says. "His background is very much Actors' Studio, and I worked with three teachers, two of whom were very much of that group, and then my husband [the late acting teacher] Charles Kakatsakis. So to some degree we were always going to work well with each other. You work with what is there in front of you and if he's pulling away from me, then I either have to go after him or stay put. And then when shooting's done Bruce turns right back into himself. He can be wicked. His sense of humour is really wild and he teases me all the time."

The newcomer on set was Will Forte, the Saturday Night Live regular who has publicly wondered how on earth he ended up in the company of these "real actors". She laughs. "He wasn't nervous on set, but, especially when it was the three of us, Bruce, myself and Will, he'd keep saying, 'What am I doing here with you two? I mean, just look at you guys!' And Bruce and I would let that go on for a while and then say, 'OK, now you can relax and be quiet now, kid.' And we'd pat his little head! But whatever Alex thought he saw in Will, Will more than acquitted himself – and the whole movie turns around him and Bruce together, so you know he got it done."

For a player of her vintage, June Squibb may have a fairly short movie and TV career, her first film having been Woody Allen's Alice in 1990, but she is a true showbiz lifer, a greasepaint gal from way back. In her depression-era hometown of Vandalia in rural Illinois, where she was born a month after the Wall Street crash, she knew she was a bit different. "It was a small midwestern town in south-eastern Illinois, not a lot different from the places in the movie. I knew all those people in Nebraska, I recognised Kate completely. I've seen scenes like those groups of utterly silent men watching the football game on TV, or moments just like that, all my life, throughout my family and friends' households there. It was a normal small town of about 5-6,000 people, and I had lots and lots of friends, I was a cheerleader and a majorette and all those things you'd expect of a girl in the 30s and 40s. But the thing I always thought of myself as was an actress. I never thought: 'I want to be.' It was always: 'I am.' I felt that from an early age, and I always thought perhaps I don't really belong here."

She fetched up in 1950 at the Cleveland Play House in Ohio doing musical theatre for several years. "And already when I was there I felt a lot more comfortable and in the right place, finally. The director there decided I was going to sing. I didn't decide – he did. So I ended up doing all the comedienne's roles in the musicals. So in New York, that's the group I fetched up with, the musical theatre crowd. New York, oh my God, in my early 20s. I felt, this is home, this is really where I belong. And I did stay there, I had my own apartment there until about two years ago, when I moved out here to California."

And she was on Broadway back when Broadway was really Broadway – Sweet Smell of Success, huge audiences, flashbulb openings, the high tide of the American stage musical. She worked with Jerome Robbins in the original cast of Gypsy alongside Ethel Merman ("this wonderful, gossipy hausfrau"), knew Steven Sondheim, sang with Robert Goulet, and danced with Kenneth MacMillan and Charles Durning – both better known later in life as splenetic movie Irishmen, but both famously light-footed as young men. In the down season she and her friends would model for paperback covers and work the big conventions. "I was the Ortho Bug-Spray girl at one of those, and you can imagine what sort of outfit it was – antennae! And I got to talk to the kind of gentlemen who are … who are interested in Ortho Bug-Spray!" – another big chuckle.

Her husband persuaded her she had more in her than song and dance and they worked to sharpen her dramatic chops in the emerging off- and off-off-Broadway realms. She was busy, sort-of-famous and respected. "I always made a decent living." Likewise, since her move to TV and movies 25 years ago, you spot her in reruns all the time, once you know who she is.

Since Alice, she's appeared in Scent of a Woman, with Philip Seymour Hoffman, whom she remembers as "just a darling boy, I called him the Teddy Bear. I told my husband, even then, that he was going to become one of the best actors you'll ever see. Everyone knew it." A small role in The Age of Innocence got her close to Martin Scorsese, whom she liked "because he really directs you – a lot of them don't. What he really wanted was reality. All of them in fact want reality. Allen was like that. The first thing he said to me was: 'I don't care what you do – just be real.' And Scorsese, the same – they're small roles for me, but they're all very worked up and thought through, and important to the directors on their terms, no matter the size of the part."

June Squibb may top out at 5ft 2in. But in the middle of her ninth decade, we're finally clocking her as an acting colossus.