Does hyperbolic geometry actually take place on a hyperboloid? Spherical geometry can be very easily seen in action on a sphere: if I drew on a ball, I would find triangles with sums of angles greater than 180 degrees, and parallel lines that intersect. If I were to draw on one of these
would I get the properties of hyperbolic geometry such as 5 squares meeting at a corner?
 A: No.
Hyperbolic geometry is geometry on a surface of constant negative Gaussian curvature. The hyperboloid has positive curvature, so the natural geometry on that surface isn't hyperbolic geometry.
There are models of the hyperbolic plane that use the hyperboloid. But they come with their own metric, their own way of how to compute angles and distances, which isn't the same as when you just measure them on a hyperboloid in 3D Euclidean space. Due to this I'm considering the hyperboloid model one of many models, all of which depict some properties quite naturally while contradicting intuition for some other properties. The tradeoffs make some models particularly well suited for particular tasks.
There is no way to get the complete infinite hyperbolic plane embedded into 3D Euclidean space. There exist several finite surfaces of constant negative curvature, such as the Tractricoid or Amsler's surface, but they don't have the right global structure. They either close in on themselves or end at a rim in places where the actual hyperbolic plane would continue. So these are local models are best, useful for small figures but misleading for the overall structure.
My PhD thesis has some illustrations of different models in section 2.1.
