As background information you need to know that an A0 sheet of paper has an area of 1 square metre and when it is folded in half its aspect ratio does not change. It then has an area of 0.5 square metres and is the size of an A1 sheet. The dimensions of an A2, A3 and A4 sheet are similarly defined by halving the area and maintaining the aspect ratio.

You can get an approximate answer by measuring an A4 sheet of paper and squaring twice but this will not give you the answer I am seeking because you will not be able to measure with sufficient accuracy.

ANSWER

First find the longer side of an A0 sheet of paper. Let this be x metres. As the area is 1 square metre, the shorter side must by 1/x metres. The aspect ratio is the longer side divided by the shorter side which is therefore x divided by 1/x which is x².

When it is folded in half to produce an A1 size sheet, shorter side of the A0 sheet becomes the longer side of the A1 sheet and the shorter side of the A1 sheet is half the longer side of the A0 sheet. So the aspect ratio of the A1 sheet is 1/x divided by x/2 which is 2/x².

The two aspect ratios must be the same so
   2/x² = x²
i.e. x⁴ = 2

The value of x (the longer side of the A0 sheet in metres) is therefore the fourth root of 2 which is approximately 1.189 metres.

The aspect ratio is x² which is the square root of 2 or approximately 1.414.

The longer side of the A4 sheet is the longer side of the A0 sheet divided by the aspect ratio four times. That is the same as dividing by 2 twice or dividing by 4. So the longer side of an A4 sheet is x/4 metres. To convert to centimetres, multiply by 100. The longer side of an A4 sheet is therefore 25x centimetres. This is the value in the display of the calculator after pressing the square root key twice so to find the original number we must square 25x twice which is (25x)⁴.
  (25x)⁴ = 25⁴ * x⁴
but we found that x is the fourth root of 2
  so x⁴ = 2.
The required answer is therefore 25⁴ * 2 = 781250. This can be calculated accurately with the simplest of electronic calculators as no fractional parts are involved.

Measuring an A4 sheet of paper in centimetres and squaring it twice does not work because you cannot measure to that accuracy nor can the sheet of paper be manufactured to that accuracy. For example 29.73 squared twice is 781231.30335441 which, when rounded to 781231 is quite a long way short of the required answer. Any other measurement to even four decimal places of accuracy (well beyond practicality) would still not yield the right answer.

I received only two correct answers, from John Stafford and John Bownas but because there was not a minimum of three, no prize was awarded this month.