  We present a physically plausible solution representing Einstein's cluster
mimicking the behaviors of compact star in the context of teleparallel
equivalent of general relativity. The Teleparallel gravity (TEGR) is an
alternative formulation of gravity which uses tetrads as the dynamical
variables. We focus on two particularly interesting scenarios. First, we
develop the Einstein clusters in TEGR field equations using effective
energy-momentum tensor for diagonal as well as off-diagonal tetrad. We then
study the clusters in modified $f(T)-$gravity for anisotropic fluid
distribution. Based on these two theories, we further study the solution
without net electric charge and then for charged solution. For charge parameter
$k \rightarrow 0$, the charged solution reduces to neutral one. Our
calculations show that when charge increases, the stiffness of the EoS also
increases. This is due to increase in adiabatic index and sound speed
approaching speed of light. When the charge increase beyond a certain limit
($0\le k \le 1.3 \times 10^{-5}$ and $0\le k \le 1 \times 10^{-6}$), the
compactness parameter crosses the Buchdahl limit i.e. $2M/R>8/9$ and the
solution start violating the causality condition. We test the
Tolman-Oppenheimer-Volkoff (TOV) limit for such compact objects. We analyze the
static stability criterion of the Einstein clusters for both charged and
uncharged case, and the stability of such compact objects is enhanced by the
presence some net electric charge. Finally, we compare our solution for pure
general relativity. As a result, we concluded that the Einstein cluster
solution do exist in pure GR, however, physically unfit to mimic compact stars.
We have also extend our findings by assuming the diagonal or off-diagonal
tetrad and specific case of $f(T)$. In such models, Einstein's cluster
solutions do exist however can't mimic the properties of a compact star.
