This paper deals with gradient Ricci solitons on Kropina measure spaces. A Kropina metric \(F\) on a measure space \((M,m)\) with the navigation data \((h, W )\) is taken. An equivalent characterization for Kropina measure space \((M, F ,m)\) to be a gradient almost Ricci soliton is given, which implies that every gradient almost Ricci soliton has vanishing \(S_{BH}\)-curvature. Based on this, it is proven that a Kropina measure space \((M, F, m)\) is a gradient almost Ricci soliton if and only if \((M,h, f )\) is a Riemannian gradient almost Ricci soliton, \(W\) is a unit Killing vector field on \(M\) and \(f\) satisfies a differential equation, where \(f\) is the potential function of the measure \(m\). As an application, some new shrinking, steady and expanding gradient Ricci solitons are constructed.