We consider the problem of fitting a centered ellipsoid to $n$ standard Gaussian random vectors in $\mathbb{R}^d$, as $n, d \to \infty$ with $n/d^2 \to α> 0$. It has been conjectured that this problem is, with high probability, satisfiable (SAT; that is, there exists an ellipsoid passing through all $n$ points) for $α< 1/4$, and unsatisfiable (UNSAT) for $α> 1/4$. In this work we give a precise analytical argument, based on the non-rigorous replica method of statistical physics, that indeed predicts a SAT/UNSAT transition at $α= 1/4$, as well as the shape of a typical fitting ellipsoid in the SAT phase (i.e., the lengths of its principal axes). Besides the replica method, our main tool is the dilute limit of extensive-rank "HCIZ integrals" of random matrix theory. We further study different explicit algorithmic constructions of the matrix characterizing the ellipsoid. In particular, we show that a procedure based on minimizing its nuclear norm yields a solution in the whole SAT phase. Finally, we characterize the SAT/UNSAT transition for ellipsoid fitting of a large class of rotationally-invariant random vectors. Our work suggests mathematically rigorous ways to analyze fitting ellipsoids to random vectors, which is the topic of a companion work.