In this work we investigate the stability and instability properties of a
class of naked singularity spacetimes. The first rigorous study of naked
singularity formation in the spherically symmetric Einstein-scalar field system
was due to Christodoulou, who constructed a family $(\overline{g}_k,
\overline{\phi}_k)$ of $k$-self-similar solutions, for any $k^2 \in
(0,\frac{1}{3})$. We extend the construction to produce examples of interior
and exterior regions of naked singularity spacetimes locally modeled on the
$(\overline{g}_k, \overline{\phi}_k)$, without requiring exact self-similarity.
The main result is a global stability statement under fine-tuned data
perturbations, for a class of naked singularity spacetimes satisfying
self-similar bounds. Given the well-known blueshift instability for suitably
regular naked singularities in the Einstein-scalar field model, we require
non-generic conditions on the data perturbations. In particular, the scalar
field perturbation along the past lightcone of the singular point $\mathcal{O}$
vanishes to high order near $\mathcal{O}$. Technical difficulties arise from
the singular behavior of the background solution, as well as regularity
considerations at the axis and past lightcone of the singularity. The interior
region is constructed via a backwards stability argument, thereby avoiding
activating the blueshift instability. The extension to the exterior region is
treated as a global existence problem to the future of $\mathcal{O}$, adapting
techniques of Rodnianski and Shlapentokh-Rothman for vacuum spacetimes.
