Walter is on the right track.
What we see above is essentually a loop-unrolled version of the reference BLAS algorithm of DNRM2. The algorithm, having much longer vectors in mind, was designed to avoid unnecessary overflow and underflow while still making just one pass through the data. Unfortunately, the reference BLAS implementation asks whether abs(x(k)) ~= 0 in the loop. Floating point equality/inequality comparisons are not welcome when the generated code must adhere, say, to the MISRA standard, and calls to NORM are quite common, so we considered how to modify the algorithm without the ~= 0 comparisons.
The number in question is the largest initial value of the scale variable that guarantees that (abs(x(k))/scale)^2 does not underflow and is not denormal for any value of abs(x(k)) > 0, i.e. (abs(x(k))/scale)^2 >= realmin. Hence, the initial value of scale is eps*sqrt(realmin).