^ But that's just the point, a 300 km/sec starship could be built by the end of this century using off the shelf technology.
A 3000 km/sec starship would require at least a fusion power source, that is 1% of the speed of light the rule here is take the distance to the star in light years and multiply it by 100 years, so a 3000 km/sec could reach Alpha Centauri in 440 years, I don't know if we'll have the technology to build it by the end of this century, possibly if we have reliable fusion by the end of this century. Its likely the fusion fuel would be deuterium and tritium same stuff to make hydrogen bombs out of, another possibility is helium-3 and deuterium, but that is harder to fuse and would require the mining of a gas giant to obtain enough Helium-3, the most likely gas giant probably would be Saturn due to its lower gravity, though I think it might be a stretch to expect mining operations in the vicinity of Saturn by 2100.
A 30,000 km/sec starship would require a staged fusion rocket or antimatter, it would be a hideously expensive thing to have throw-away rocket parts, probably only a massive government program could attempt this, another wild possibility is a giant laser to push a starship up to 10% of the speed of light, to calculate travel time, simply multiply the distance of the star in light years by 10 years, for a trip to Alpha Centauri it would take 44 years.
I think in all cases, we need a 500 meter habitat sphere even for starships that reach 10% of the speed of light, as 44 years is half a human lifetime they'll need somewhere comfortable to live for 44 years. I think acceleration should be limited to 1% of Earth gravity max, as where talking about a rotating sphere.
1% of Earth's gravity is 10 cm per second squared, and at that rate it would take 9.51 years to reach 10% of the speed of light or 30,000 km/sec.
At 0.1% of Earth gravity or 1 cm per second squared it would take 9.51 years to reach 1% of the speed of light or 3000 km/sec.
At 0.01% of Earth gravity or 1 mm per second squared it would take 9.51 years to reach 0.1% of the speed of light or 300 km/sec.
If we wait for these more advanced technologies, we are allowing ourselves more time to destroy ourselves. If the goal is to preserve the human race, there really is no hurry to get there, but there is a hurry to launch a starship, the sooner we launch this ship the sooner we would have "bought" the insurance policy against the extinction of the human race. I think it would be easier to develop artificial intelligence than fusion reactors or antimatter reactors and a way of mass producing antimatter in a large enough scale to power a starship, and if we could do that, how much easier would it be to destroy ourselves with control over energies like that? I would feel safer if we launched as starship sooner rather than waiting for the technologies to arrive to send a human crew over there within their own lifetimes.