Can we build a Starship with our current technology in the next 10 years?
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Remember that the only condition for a star ship, is to be able to have thrust all the time. Even a very small thrust is enough to reach relativistic speeds, and therefore make the travel-time human-scale. FOR THE TRAVELERS!
Now, the problem is that unless the travelers include the trillionaire that will finance the projects, nobody will want to finance a travel, where the travellers reach the destination after their dead, and can give back news only after the payers' children are dead.
But travelling one year on board to reach Proxima Centauri (4 light years away) wouldn't even be tiring for the traveller.
(Ok, perhaps we could pay for this destination, after all, we'd get news about landing ten years later).
There are two limitations you did not take into account. The first is the rocket equation, https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation . You need a thrust with a very high exhaust velocity - much higher than what an ion drive can produce - to get to relativistic speeds. Otherwise something like 99.999999999999% of the ship's mass will need to be propellant. I worked it out a few weeks ago at https://news.ycombinator.com/item?id=12281830#12282399 . The ratio of payload to propellant for ~0.8c using engines 10x better than an ion engine was 1 to 74 quadrillion.
The second is to have enough thrust to get to relativistic speeds for the human passengers in their lifetime. A engine with an ISP of 8000 s (about 10x better than the current best ion engine) would get there eventually, but long after any crew died of old age.
To get 4.25 lightyears in a year requires a continuous acceleration of a good sized fraction of a gravity. We simply don't know how to do that, outside of theoretical proposals like Dyson's "Super" Orion. Quoting https://en.wikipedia.org/wiki/Project_Orion_(nuclear_propuls... :
> At 0.1c, Orion thermonuclear starships would require a flight time of at least 44 years to reach Alpha Centauri, not counting time needed to reach that speed (about 36 days at constant acceleration of 1g or 9.8 m/s2). At 0.1c, an Orion starship would require 100 years to travel 10 light years.
Alpha Centauri is 4.37 light years away. That trip would take about 77,000 years, were Voyager 1 pointed the right direction.