There's a slight issue I believe with the other answers. If we consider moduli space as an orbifold (of complex dimension $3g-3$), and the hyperelliptic locus an immersed suborbifold (of complex dimension $2g-1$ or so), then we may (essentially) identify the hyperelliptic locus with the orbifold of $2g+2$ points on $S^2$, obtained by quotienting each Riemann surface by the hyperelliptic involution. However, how does one know that this space doesn't "cross" itself? Imagine by analogy an immersed geodesic curve on a hyperbolic surface, such that each complementary component is a disk: the preimage in the universal cover is connected, when taken as a union of geodesics, even though each geodesic lift is embedded.

This sort of crossing does not occur for the hyperelliptic locus. If two branches of the hyperelliptic universal cover in Teichmuller space were to intersect, then there would be a single Riemann surface fixed by two distinct hyperelliptic involutions. But a hyperelliptic involution fixes precisely the $2g+2$ Weierstrauss points of the surface, and is therefore uniquely determined, a contradiction. So in fact the hyperelliptic locus is "embedded", in the sense that each lift corresponds to a fixed set of a hyperelliptic involution, and distinct hyperelliptic involutions give distinct components in Teichmuller space.