Here on the eleventh floor she is physically and metaphorically at the top of the building. Ms Strachan, at 53, has achieved what no woman has before, heading one of the oldest Whitehall departments with a staff of 26,000.
She stepped into the job on 1 March, succeeding Sir Brian Unwin, who has gone to the European Investment Bank. As the third of John Major's much-trumpeted women appointees to high office - following in the footsteps of Barbara Mills as Director of Public Prosecutions and Stella Rimington at MI5 - she has attracted keen media interest.
'I don't think anyone ever interviewed Sir Brian,' a press officer remarked.
Ms Strachan has been with Customs & Excise for more than 30 years, joining straight from a politics degree at Manchester University. She will not admit to expecting the promotion to chairman, even when the call came at the end of last year from the head of the Home Civil Service, Sir Robin Butler. But when offered the job she was 'absolutely delighted'.
For the previous five years she had been one of the department's deputy chairmen, heavily involved in negotiations for the single European market. Her work on this contributed to journalists in the national media awarding her the UK Woman of Europe award last year.
Now in the hot seat, she says she can bring a new approach to the job despite her long history with the department. 'I know the department very well, but its work has been changing throughout the time I've been here. There's never a feeling that you know it all.'
Her main challenge will be to manage change, she said. 'The single market means we have to change the way we deal with customs, excise, VAT, and collect statistics. That means that most people in the department have to do their jobs in a different way, quite often in a different place from where they were doing it before. It requires quite sensitive managing.' It has also meant redeployment and redundancies, although the department says that the single market resulted in only six compulsory redundancies.
Ms Strachan said that being able to move staff between functions has kept job losses low; when activity switched from being port-based to inland, about 1,600 staff were moved.
Another priority is the department's image. Although the Matrix Churchill affair is off-limits - 'That is all in front of Justice Scott and I ought not to be in any way pre-empting his inquiry' - she is keen to play down notions of heavy-handed VAT men.
'With VAT you're talking about tax, and nobody likes to be taxed.' The department receives 'few' complaints: 'Most of them are not about the conduct of officers but about the very rigid VAT penalty regime.' The fact that this was eased in the Budget will come as a relief to VAT officers, she believes. 'They will be delighted by it because they had felt they were trying to operate a regime which was too rigid.'
Customs & Excise makes great efforts to talk to businesses, she argues. 'The fact that we're a tax-collecting department means we'll always have to work at our relationship with businesses and with trade associations. You may find it surprising but that does come very naturally to us - we just have to make sure we keep in touch with business concerns.'
Ms Strachan is unfailingly polite, smiles a great deal and answers questions methodically, with a civil servant's practised discretion. But she did not always see her career in Whitehall. 'In my last year at university I hadn't the remotest idea of what I wanted to do. I sent applications off in all directions, including Marks & Spencer and advertising agencies.' It was the selection process for the Civil Service, analysing problems and discussing solutions, that swayed her. 'I went through it and I thought, 'this is interesting, I could enjoy this'.'
And so it has proved. Although her first choice was the Department of Education, she landed up in Customs & Excise, where she has stayed, with occasional forays on secondment to the Treasury and the Home Office.
One of her spells outside the department, in the mid-1980s, was in the Joint Management Unit, the Treasury/Cabinet Office 'think tank' that aimed to push through management reform in the Civil Service. Part of its legacy are the Next Steps agencies, still being created. She disagrees that reform to improve management accountability is slow.
'It's a continuing process: quite often you're trying to apply quantification to things which are not easy to quantify.' With VAT, for example, the department can measure the costs of collection, but it becomes more difficult to measure activities such as improving compliance. 'We've got researchers who've been developing better indicators of our effectiveness. It's not that it's slow, but that it's difficult.'
In fact, Customs & Excise has always been one of the more advanced departments. Its 1991/2 annual report, bristling with coloured tables and graphics, includes running costs for its 21 executive units, set up in April 1991 as part of the Next Steps programme. The units, which now cover around 95 per cent of the staff, have delegated authority over finance and personnel management and publish their own reports.
'Customs & Excise has been looking at bottom lines of various kinds for years,' she explains. 'We were in very early with the Financial Management Initiative (to improve financial reporting) in the 1980s, setting out our priorities, aims and objectives and quantifying them. We've been producing quite sharply focused management plans and annual reports for some years now.'
While welcoming this kind of accountability in the public sector, she is clear that a line has to be drawn. 'After you've applied all the private sector disciplines, you then have to apply all the public sector disciplines, such as accountability and propriety.' It would be possible, she agrees, to become bogged down in it. 'I'm quite clear about what our job is. It isn't to apply management theory - it's to collect revenue and protect society from importation of drugs, arms and ammunition.'
Has she never been tempted to work in the private sector? Discreet pause. 'I remember once, back in the early Seventies, when I was feeling faintly fed up with what I was doing, thinking perhaps I should make a change. But before I was able to do anything about it, they promoted me into an interesting job.'
Only something 'really interesting and different' would have a chance of luring her away.