Fractional continuum models provide a natural extension of classical elasticity by incorporating nonlocal effects that arise from material microstructure and finite-range interactions. In particular, formulations based on fractional derivatives allow strain measures to depend not only on local displacements but also on their spatial neighborhood, offering a realistic description of size-dependent and anomalous mechanical responses. While noninteger formulations have been widely explored in continuum and structural mechanics, variational approaches remain comparatively less common in the modeling of nonlocal elastic behavior. Accordingly, in this work, the axial deformation of a one-dimensional elastic bar is investigated within a fractional variational framework. Starting from an energy-based formulation, a generalized Euler–Lagrange equation governing the displacement field is derived. The resulting model consistently recovers the classical elasticity equation in the appropriate limit, while capturing size-dependent nonlocal effects. The governing fractional differential equation is reformulated in an equivalent integral form, which allows the construction of explicit analytical solutions and provides a direct basis for their numerical evaluation. This combined analytical-numerical formulation enables a systematic study of how nonlocality influences the axial response of elastic structures. The results demonstrate that the proposed model provides a smooth and physically consistent transition between classical local elasticity and nonlocal behavior, offering a robust tool for modeling size-dependent and microstructure-driven effects in axial deformation problems.