Is the inside of a sphere a hyperbolical surface? So since an elliptic surface with constant curvature would be a sphere, would a an hyperbolical surface with a constant curvature be the inside of a sphere if we were to go out from inside the sphere? Many thanks.

No, from the point of view of a surface, there is no distinction between "the surface from the inside" and "the surface from the outside.

The model surface with constant negative curvature (and hence the hyperbolic analogue of the $2$-sphere) is the hyperbolic plane. A theorem of Hilbert states that, unlike the $2$-sphere, the hyperbolic plane cannot be embedded isometrically into $\Bbb R^3$. One _can_ still find a surface in $\Bbb R^3$ that has negative constant curvature (i.e., embed the hyperbolic plane locally), e.g., half of a tracticoid. Unlike the hyperbolic plane, any such space is incomplete.