  Enflo constructed a countable metric space that may not be uniformly embedded
into any metric space of positive generalized roundness. Dranishnikov, Gong,
Lafforgue and Yu modified Enflo's example to construct a locally finite metric
space that may not be coarsely embedded into any Hilbert space. In this paper
we meld these two examples into one simpler construction. The outcome is a
locally finite metric space $(\mathfrak{Z}, \zeta)$ which is strongly non
embeddable in the sense that it may not be embedded uniformly or coarsely into
any metric space of non zero generalized roundness. Moreover, we show that both
types of embedding may be obstructed by a common recursive principle. It
follows from our construction that any metric space which is Lipschitz
universal for all locally finite metric spaces may not be embedded uniformly or
coarsely into any metric space of non zero generalized roundness. Our
construction is then adapted to show that the group
$\mathbb{Z}_\omega=\bigoplus_{\aleph_0}\mathbb{Z}$ admits a Cayley graph which
may not be coarsely embedded into any metric space of non zero generalized
roundness. Finally, for each $p \geq 0$ and each locally finite metric space
$(Z,d)$, we prove the existence of a Lipschitz injection $f : Z \to \ell_{p}$.
