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An atom placed inside a cavity of finite dimension offers many interesting features, and thus has been a topic of great current activity. This work proposes a density functional approach to pursue both ground and excited states of a multi-electron atom under a spherically impenetrable enclosure. The radial Kohn-Sham (KS) equation has been solved by invoking a physically motivated work-function-based exchange potential, which offers near-Hartree-Fock-quality results. Accurate numerical eigenfunctions and eigenvalues are obtained through a generalized pseudospectral method (GPS) fulfilling the Dirichlet boundary condition. Two correlation functionals, \emph{viz.,} (i) simple, parametrized local Wigner-type, and (ii) gradient- and Laplacian-dependent non-local Lee-Yang-Parr (LYP) functionals are adopted to analyze the electron correlation effects. Preliminary exploratory results are offered for ground states of He-isoelectronic series ($Z=2-4$), as well as Li and Be atom. Several low-lying singly excited states of He atom are also reported. These are compared with available literature results -- which offers excellent agreement. Radial densities as well as expectation values are also provided. The performance of correlation energy functionals are discussed critically. In essence, this presents a simple, accurate scheme for studying atomic systems inside a \emph{hard} spherical box within the rubric of KS density functional theory.