  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]
