Quantitatively correlating the amorphous structure in metallic glasses (MGs) with their physical properties has been a long-sought goal. Here we introduce 'flexibility volume' as a universal indicator, to bridge the structural state the MG is in with its properties, on both atomic and macroscopic levels. The flexibility volume combines static atomic volume with dynamics information via atomic vibrations that probe local configurational space and interaction between neighbouring atoms. We demonstrate that flexibility volume is a physically appropriate parameter that can quantitatively predict the shear modulus, which is at the heart of many key properties of MGs. Moreover, the new parameter correlates strongly with atomic packing topology, and also with the activation energy for thermally activated relaxation and the propensity for stress-driven shear transformations. These correlations are expected to be robust across a very wide range of MG compositions, processing conditions and length scales.