You don't need to specify the potential beyond the Dirichlet boundaries:
L = 2;
V[x_] = 0; (* will only be used in [-L,+L] *)
{vals, funs} =
NDEigensystem[{-Laplacian[u[x], {x}] + V[x]*u[x],
DirichletCondition[u[x] == 0, True]},
u[x], {x, -L, L}, 10,
Method -> {"SpatialDiscretization" -> {"FiniteElement", {"MeshOptions" -> {MaxCellMeasure -> 0.01}}}}]
The eigenvalues are correct numerically:
vals - Table[(n*π/4)^2, {n, 10}]
(* {4.27502*10^-12, 2.12463*10^-10, 2.37989*10^-9,
1.33551*10^-8, 5.09357*10^-8, 1.5208*10^-7,
3.83468*10^-7, 8.54395*10^-7, 1.732*10^-6,
3.25887*10^-6} *)