A hypertree, or $\mathbb{Q}$-acyclic complex, is a higher-dimensional analogue of a tree. We study random $2$-dimensional hypertrees according to the determinantal measure suggested by Lyons. We are especially interested in their topological and geometric properties. We show that with high probability, a random $2$-dimensional hypertree $T$ is apsherical, i.e. that it has a contractible universal cover. We also show that with high probability the fundamental group $π_1(T)$ is hyperbolic and has cohomological dimension $2$.