Here's today's question.

If the house of Annie Lennox was an actual house and not just an exhibition at the National Portrait Gallery of Scotland, what kind of house would it be? A rambling Victorian mansion full of dusty little corners under the stairs? An ultra-modernist duplex containing nothing made before 1982? A Barratt home in need of TLC? A Bibaesque boudoir perfect for pop star fantasies or a global kitchen perfect for a cosmopolitan activist?

Actually I'm thinking this notional house might be a bit more like Terminal Five at Heathrow. And if you ask Annie Lennox she might even agree. "People ask: 'Where do you call home?' Because I'm such an itinerant traveller. I've lived in London for so many years and I do now have a fondness for it," she says. "It took me many, many years to feel connected because it's a huge metropolis. I would come back and leave, come back and leave. Because my kids were brought up there I've got a different relationship to it, but I still don't know if it's home."

We are sitting in the portrait gallery surrounded by images of Annie the singer, Annie the activist, Annie the pop star, Annie the magazine cover star, Annie the clothes horse. But I'm talking to Annie the woman; a chatty, laidback, often amused ("Are you having a senior moment?" she asks when I lose my train of thought at one point and later, when she struggles to think of the word she wants, admitting she's having one too) and, by her own admission, a really rather loved-up woman. Home turns out to be where the heart is.

"For me home is where my husband is. When we're together, that's home. It's less of a bricks and mortar thing, more a presence." Lennox married Dr Mitch Besser, a South African gynaecologist and philanthropist, last September. Her two adult daughters Lola and Tali were among the bridesmaids.

Third time lucky. After two marriages that didn't go the distance Lennox, now 58, feels the foundations are firmer this time around. "He's my counterpart. I've finally, finally met the man.

"We're well suited, well matched. There's that compatibility. Compatibility is the key. I think when you're younger you're drawn magnetically towards whoever you're drawn towards for all sorts of reasons, but they're very subliminal. You don't know why you're drawn and as you get older you start to maybe have to question more deeply: 'Why am I repeating this pattern of behaviour?'

"Now that I'm coming from a place in my life that is very peaceful and very content and very happy in a relationship, and I'm so glad, and that's really what I always wanted."

Love is no longer a stranger.

Before I speak to her I'd come up with the theory that Annie Lennox has been around long enough now for us to have forgotten how good she is. It's been a few years since she released an album and recently she has been far more visible as an activist and HIV campaigner. But she still sees herself as a singer. She's quite good at it, to be fair. Even if you weren't keen on her 1990s plushly upholstered, rather Habitaty output, it's worth recalling how original and shocking those early Eurythmics records were. Even today's bands are fulsome in their praise. Glasgow's great white hopes Chvrches recently told me, "We love Annie Lennox. She's been a massive influence."

And yet for a long time Lennox's success seemed unlikely. She left Aberdeen at 17 to study music in London only to hate it straight away. "I looked around and I realised I was in the wrong place. 'This isn't right.' I just lived on the grant and I had little part-time jobs in bookstores and supermarkets and bars and I was troubled because I really thought: 'It's a dead-end. What am I going to do? Your life is over.' All the dire premonitions that my parents had constantly made – bless them, they meant well, but they were always coming from that very threatening place. 'If you don't stick in at school you'll end up in a factory.' That was constantly being said and I understand why because they had to work so damn hard."

She'd gone to London with a vague notion of joining a classical ensemble. But then she discovered a friend's record collection and decided this would be her new direction. "It was like the world opened up to me. I thought, 'I can join a band. I can sing.'"

She started trawling the back pages of the Melody Maker, joined three dreadful bands and then got introduced to Dave Stewart. "We were hopeless, the pair of us. But it started from there."

It was a bit of a false start. They became lovers and partners, formed The Tourists, made some hit singles and three albums and were hated by the music press. When the band split the two of them were left with huge debts. Tough lessons had to be learned. "When we started with The Tourists we were going in a certain direction and the record company we'd signed to didn't want us to do that. So we were in litigation for a whole year and we were caught. When you're young, you're ambitious, and we couldn't do anything for a whole year. We could do nothing. We had to sign on the dole which I hated because I was very proud and I didn't want to feel I was just taking something. I always liked to work but I couldn't work because I was in a band and I had to work on being in a band. That was a tough year. You learn so many pitfalls in the industry. We signed that stupid contract in perpetuity. You think 'Jesus, how did that happen? Why did we do that?'"

Lennox's relationship with Stewart didn't survive in the heat of all this. By far the most poignant items in the portrait gallery are some of the smallest – a series of self-portraits Lennox did at the beginning of the eighties that have not been seen in public before. "I was terribly depressed and lonely that day," she says in the accompanying wall panel. I'd read she had suffered a nervous breakdown. But she says that's not so.

"I never actually had a breakdown. I was never under any medication. I was never under any doctor's orders or put in an institution. If you read some of the things that are written you would think I was in a straitjacket, barking mad, taken away with men in white coats.

"In a sense it was one of those struggles where things break down and part of you is broken, so it was very hard. Dave and I were left with the debts of the whole band. We'd made money for a lot of people and at the end of the day we had debt after three albums. It felt such a struggle to survive from a practical point of view.

"Dave and I were a couple and we split up and that was very painful but we knew we had to be together as a musical partnership so I mean it was just - raw." You can hear the scour in her voice when she says that word. "It was raw," she continues, "because we were each other's best shot. I didn't want to work with anyone else. He didn't want to work with anyone else. So we became this duo and we were very, very focused."

Maybe then it's a case of being strongest at the broken places because two years after The Tourists split, Eurythmics were one of the biggest bands in the world. Stewart got into the new electronics, Lennox began to write lyrics and the pair decamped to West Germany to work with krautrock guru Conny Plank, who told Lennox to sing like she was washing the dishes. Fortunately she ignored him. They made one album, In The Garden. Then they made another one, Sweet Dreams, and the accompanying videos – which saw Lennox play around with androgyny and sexuality – changed everything. "We ended up skyrocketing with that album," she says. "We weren't totally obscure, but we were catapulted into massive fame after four albums. It was like we'd been hammering on this door for a long time and finally the door opened and it was like: 'OK, you've got what you want, come on in, here you go, have your world blown open.'

"Then it was a decade of intense creativity. There was nothing more important than Eurythmics for us. Nothing. In a way it was a great opportunity to be creative, but at the same time, personally, it was restrictive and strange. I was on another planet. I wasn't with Dave any more, but I was with him all the time. And life on the road is tough, there's no question about that. It's really tough and I was unhappy in it. But you're just on a mission."

It's tempting to ascribe her first marriage – a short-lived one to German Hare Krishna devotee Radha Ramen – to that unhappiness. But that's just me projecting. She does say that being famous – and by the mid-eighties she was very, very famous – didn't make things easy in terms of relationships. She did marry again, in 1988, to film and record producer Uri Fruchtmann, the father of her daughters. They lasted 13 years before separating in 2000, by which time Lennox was the elder stateswoman of pop.

If anything, she's a citizen of the world now, she'd say. Success has taken her far from home in a way, far from the background she came from in Aberdeen. Last year she filmed an episode of the BBC genealogy series Who Do You Think You Are? She explored her roots and found the "invisible threads" that trail behind her. "And it all went back to poverty. It was deeply moving. People's lives were really, really tough. They had to face illness, they had to face arduous hard work and cruel conditions. The word 'pauper' came up. Or they would have to go with an illegitimate child to the church elders and make their case and they were punished. My generation – post-Second World War – we're the luckiest bastards on the planet."

You can trace her activism to her family. On her father's side stretches back a long line of trade unionists, Labour Party members, Communist Party members too, until the Russian invasion of Czechoslovakia in 1968 anyway. "Then they were just like, 'Oh, this is not what we paid into.'"

And yet Lennox herself doesn't vote. "I'm apolitical. You could say that's irresponsible but I think if everyone didn't vote they'd have to have a look, so in a way not voting is my choice too. I don't want to vote for anyone I don't totally believe in."

When she says she's not political what she really means is she's not party political. Because everything's political, isn't it? Even celebrity. "We never used that word. I never knew that word until the nineties. I don't know what the hell that is. It's a generic term for anybody from a serial rapist to - whatever." As she speaks, she finds her voice. "What is celebrity? It's something so false, it's totally false. It feeds off itself. It's a consumption. It's the exhibitionist and the voyeur, it's the frenzy and the currency. It's the buying-in and the cashing-in of corporate institutions that basically have something to sell; a lifestyle, an image, a brand, and in a way we lose a lot of our humanity by that branding. A brand is a huge anonymous thing and behind that we cease to be human beings. It's just like we're being sold. That's how I feel about it."

Where have we arrived? Somewhere far from home. Annie Lennox gets up and leaves the building. n

The House Of Annie Lennox continues at the National Portrait Gallery of Scotland, Edinburgh, until June 30.