Sometimes, even now, Arctic Monkeys are surprised by their own bigness. When their tour bus pulled into Malahide, a few miles outside Dublin, Jamie Cook, the band’s guitarist, looked up from his PlayStation game to see 250 acres of parkland in the grounds of a stately home, ringed by newly erected fences and dotted with tents.

A small army of gardai, vendors, security guards and stage riggers were working away. There were impressive backstage facilities (comfortable dressing rooms, drinking patio, top catering) both for the headline artists and for their personally chosen support acts, the Coral and Supergrass – two older, more experienced bands that the four young schoolmates from Sheffield had been inspired by.

Over two nights, 26,000 Arctic Monkeys fans would fill this space. “F***ing hell,” said Cook. All this industry and organisation, just for them. “That’s a bit weird.” Drummer Matt Helders said his dad, over from Sheffield to see his son perform, was equally bewildered by the spectacle of “all these people, working ’cause of you”. “My dad wanted me to try and kick one of them out or summat,” said Helders with a grin. That was a joke, obviously. No Arctic Monkey would dream of being so impolite, far less of pulling a rock star strop, even if they have sold 3.5 million albums in fewer than two years.

The band were kicking off a run of big summer shows that would take them all over Europe. In April they had put out their second album, Favourite Worst Nightmare: the fastest-selling album of the year so far, and the quick-smart follow-up to Whatever People Say I Am, That’s What I’m Not, itself the fastest-selling debut album ever and winner of two Brit Awards and of the 2006 Mercury Music Prize. The second album has now also been nominated for the Mercury Prize – making Arctic Monkeys the first band to be nominated again the year after winning it – and its release was accompanied by sell-out indoor shows in the UK, European dates and an American tour. Now they were stepping up a gear with prestigious headline appearances at the Glastonbury and T In the Park festivals.

At 8.48pm on the Sunday night, the stage lights dimmed and the Malahide crowd roared. Someone dressed in a full Spider-Man costume sauntered in from the wings. The crowd kept roaring. It was Helders (his bandmates, in their broad Yorkshire accents, also call him by his surname). He was followed by 22-year-old Cook (“Cookie”), bass player Nick O’Malley (“Malley”), 22, and lastly, the boyish figure of singer and songwriter Alex Turner (“Al”). Like Helders, he’s 21.

Arctic Monkeys launched with ferocity into The View from the Afternoon. They barely let up for the next 80 minutes. No fancy lights or stage set. Just huge, singalong songs. The crowd lustily accompanied them, even singing some of the band’s signature guitar riffs.

Afterwards, Arctic Monkeys and some family members stood outside their Portakabin dressing rooms, quietly drinking beer. Why had Helders worn the Spider-Man costume? “I dunno,” he shrugged. “It were in me case, ’cause someone got me it for me birthday.”

“Malley was like, ‘Go on, wear it…’” continued Turner. “Then we just hatched a plan. We love owt like that, confusing people.”

Arctic Monkeys are not like other bands. Not just because they have enjoyed unprecedented levels of success in the short time since I Bet You Look Good on the Dancefloor crashed into the British singles charts at number one in October 2005. And certainly not because, as popular lore has it, they were the first band to benefit from their fans sharing demos of their songs via the internet.

They’re different because, as Alex Turner said, they like confusing people: they chose not to attend the Brit Awards ceremony earlier this year, but filmed two acceptance speeches in which they dressed like the cast of The Wizard of Oz and members of Village People. Appearing on Jonathan Ross’s BBC TV show the other Friday to promote their single, Fluorescent Adolescent, they wore clown costumes, a reference to the video for the single. Yet they still refuse to take up most of the offers of exposure – magazine cover shoots and the like – that come their way.

They’re different too because they don’t like doing interviews. Really don’t like doing interviews. This story has taken the best part of a year to set up. When you do eventually get to stick a tape recorder in front of them, Arctic Monkeys clam up. Sentences begin, then peter out. Turner, so literate and perceptive in his lyrics, is tongue-tied and often inarticulate when questioned directly.

Ask them, for example, what countries they’ve enjoyed touring and the singer rejoins: “I still don’t feel like we’re sorta like… professionals. We haven’t really done that much really.”

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