Can there be a segment on a hyperbolic plane that goes from point $(0,0)$ to $(0,0)$ in the hyperbolic plane. There are some rules, though for this to work:
1) The segment must apply to the rules of segments in hyperbolic space.
2) The segment a length $\gt 0$
3) Only one single point $(0,0)$ that exists on the plane.
No, because the axioms of intermediacy are true on the hyperbolic plane. That is, if for three distinct points $A,B,C$, we say that $B$ is between $A$ and $C$ then we say that $C$ is not between $A$ and $B$. A segment that starts from $A$ and ends at $A$ will be a circle like configuration on which the intermediacy axioms do not hold. See the illustration below.
The axioms of intermediacy are the basic axioms of the so called ordered geometries. The hyperbolic plane (space) is a specialization of the ordered plane (space) space.