The medical technology for human stasis doesn't seem particularly unattainable, when compared against building a carbon nanotube tether into orbit. Some mammals are able to hibernate for months.
General anesthesia kills 1 in 100,000 patients, and that's with medical personnel standing by in actual hospitals, much less a skeleton crew of people strapped to a spacebound elevator.
We don't understand ourselves or our bodies anywhere near as well as we can understand or advance materials science. My money is on new cables before reliable stasis.
No, we don't even know if warp drives are allowed by the laws of physics. We have a pretty good idea how to do hibernation: Saturate the tissue with some cryoprotectant and freeze it very quickly. The problem is that we don't have a cryoprotectant that is sufficiently non-toxic and we can't freeze anything as big as a human quickly enough.
Yeah and since nobody knows whether the exotic matter needed actually exists, we don't know whether bending spacetime in the required way is physically possible.
On the other hand, we can successfully freeze and thaw bunny kidneys.
