The uncertainties in neutron star (NS) radii and crust properties due to our limited knowledge of the equation of state (EOS) are quantitatively analysed. We first demonstrate the importance of a unified microscopic description for the different baryonic densities of the star. If the pressure functional is obtained matching a crust and a core EOS based on models with different properties at nuclear matter saturation, the uncertainties can be as large as $sim 30%$ for the crust thickness and $4%$ for the radius. Necessary conditions for causal and thermodynamically consistent matchings between the core and the crust are formulated and their consequences examined. A large set of unified EOS for purely nucleonic matter is obtained based on 24 Skyrme interactions and 9 relativistic mean-field nuclear parametrizations. In addition, for relativistic models 17 EOS including a transition to hyperonic matter at high density are presented. All these EOS have in common the property of describing a $2;M_odot$ star and of being causal within stable NS. A span of $sim 3$ km and $sim 4$ km is obtained for the radius of, respectively, $1.0;M_odot$ and $2.0;M_odot$ star. Applying a set of nine further constraints from experiment and ab-initio calculations the uncertainty is reduced to $sim 1$ km and $2$ km, respectively. These residual uncertainties reflect lack of constraints at large densities and insufficient information on the density dependence of the EOS near the nuclear matter saturation point. The most important parameter to be constrained is shown to be the symmetry energy slope $L$ which exhibits a linear correlation with the stellar radius, particularly for masses $sim 1.0;M_odot$. Potential constraints on $L$, the NS radius and the EOS from observations of thermal states of NS are also discussed. [Abriged]
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