  The 3-body problem in $\mathbb{R}^4$ has 24 dimensions and is invariant under
translations and rotations. We do the full symplectic symmetry reduction and
obtain a reduced Hamiltonian in local symplectic coordinates on a reduced phase
space with 8 dimensions. The Hamiltonian depends on two parameters $\mu_1 >
\mu_2 \ge 0$, related to the conserved angular momentum. The limit $\mu_2 \to
0$ corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian
has relative equilibria that are local minima and hence Lyapunov stable when
$\mu_2$ is sufficiently small. This proves the existence of balls of initial
conditions of full dimension that do not contain any orbits that are unbounded.
