Modern Two in Edinburgh, which is part of the National Galleries of Scotland estate, has shut to the public for the rest of the year unless by appointment.
Gallery bosses said the combined impact of soaring inflation, the cost-of-living crisis, rising energy bills and lower visitor numbers had created the perfect storm.
More than 40 staff had already left in a 'voluntary exit' scheme to cut costs.
Sir John said his organisation was facing a funding challenge 'the like of which I have never before witnessed or, indeed, imagined' He said his priority for next year and beyond was 'how we protect the collection, keep the lights on and doors open - and that is it'.
Sir John said: "We face a funding challenge the like of which I've never before witnessed or imagined.
"Already, before the events of recent months, we were looking at a pretty substantial deficit in our budget for next year, widening in the years beyond.
"Then when you layer in the lingering impact of the pandemic, when you layer in the dramatic inflationary costs that we're seeing at the moment, the pressure to try and keep paying a fair wage to our staff and, of course, the energy costs, which for my organisation are predicted to at least double next year from a six-figure sum to a seven-figure sum.
"Then you are talking about a crisis that feels more serious and more difficult to deal with than the pandemic itself."
Created in 1999 Modern Two is home to work by Leith-born sculptor Eduardo Paolozzi which is on permanent display.
The Stairwell Project, a large- scale, permanent work by 2009 Turner Prize- winner Richard Wright is also housed there.
A spokesperson for the National Galleries of Scotland said: "The culture sector was badly affected during the pandemic and, like so many national and local cultural organisations, National Galleries of Scotland is still in a period of recovery with visitor numbers and self-generated income still well below pre-COVID levels.
"On top of the lingering impact of the pandemic, we now have a cost of living crisis with rampant inflation and the prospect of dramatic increases in energy costs. "Giving evidence at a recent meeting of the Scottish Parliament's Constitution, Europe, External Affairs and Culture Committee, Sir John Leighton, Director-General of the National Galleries of Scotland set out some of these challenges for MSPs and gave his views on some of the possible impacts on levels of service in future years.
"It should be stressed that these were potential future scenarios and National Galleries of Scotland will of course work hard, with the support of Scottish Government, to do everything possible to maintain its ambitions and position as a flagship of Scottish culture in the years to come.
"Modern Two closed as planned at the end of the Barbara Hepworth exhibition.
"We will be making announcements about the forthcoming exhibition programme for Modern Two in due course."