Scaled momentum distributions for the strange hadrons K0s and Lambda/bar Lambda were measured in deep inelastic ep scattering with the ZEUS detector at HERA using an integrated luminosity of $330 pb^{-1}$. The evolution of these distributions with the photon virtuality, $Q^2$, was studied in the kinematic region $10<Q^2<40000 GeV^2$ and $0.001<x<0.75$, where $x$ is the Bjorken scaling variable. Clear scaling violations are observed. Predictions based on different approaches to fragmentation were compared to the measurements. Leading-logarithm parton-shower Monte Carlo calculations interfaced to the Lund string fragmentation model describe the data reasonably well in the whole range measured. Next-to-leading-order QCD calculations based on fragmentation functions, FFs, extracted from $e^+e^-$ data alone, fail to describe the measurements. The calculations based on FFs extracted from a global analysis including $e^+e^-$, $ep$ and $pp$ data give an improved description. The measurements presented in this paper have the potential to further constrain the FFs of quarks, anti-quarks and gluons yielding $K^0_{S}$ and $\Lambda/\bar{ \Lambda}$ strange hadrons.