We study streaming algorithms for the fundamental geometric problem of computing the cost of the Euclidean Minimum Spanning Tree (MST) on an $n$-point set $X \subset \mathbb{R}^d$. In the streaming model, the points in $X$ can be added and removed arbitrarily, and the goal is to maintain an approximation in small space. In low dimensions, $(1+ε)$ approximations are possible in sublinear space [Frahling, Indyk, Sohler, SoCG '05]. However, for high dimensional spaces the best known approximation for this problem was $\tilde{O}(\log n)$, due to [Chen, Jayaram, Levi, Waingarten, STOC '22], improving on the prior $O(\log^2 n)$ bound due to [Indyk, STOC '04] and [Andoni, Indyk, Krauthgamer, SODA '08]. In this paper, we break the logarithmic barrier, and give the first constant factor sublinear space approximation to Euclidean MST. For any $ε\geq 1$, our algorithm achieves an $\tilde{O}(ε^{-2})$ approximation in $n^{O(ε)}$ space. We complement this by proving that any single pass algorithm which obtains a better than $1.10$-approximation must use $Ω(\sqrt{n})$ space, demonstrating that $(1+ε)$ approximations are not possible in high-dimensions, and that our algorithm is tight up to a constant. Nevertheless, we demonstrate that $(1+ε)$ approximations are possible in sublinear space with $O(1/ε)$ passes over the stream. More generally, for any $α\geq 2$, we give a $α$-pass streaming algorithm which achieves a $(1+O(\frac{\log α+ 1}{ αε}))$ approximation in $n^{O(ε)} d^{O(1)}$ space. Our streaming algorithms are linear sketches, and therefore extend to the massively-parallel computation model (MPC). Thus, our results imply the first $(1+ε)$-approximation to Euclidean MST in a constant number of rounds in the MPC model.