It seems that I have the needed example, but I want it to be simple and self-explaining...
Construct a nontrivial complete metric space $X$ with intrinsic metric which has no nontrivial minimizing geodesics.
A metric $d$ is called intrinsic if for any two points $x$, $y$ and any $\epsilon>0$ there is an $\epsilon$-midpoint $z$; i.e. $d(x,z),d(x,y)<\tfrac12 d(x,y)+\epsilon$.
A minimizing geodesic is nontrivial if it connects two distinct points.
A meric space is nontrivial if it contains two distinct points.
- Clearly, $X$ can not be locally compact.