Could we build a starship using current technology?

[David Johansen]

No

But, that's assuming something new and groundbreaking doesn't come along in the next couple years.

Highly unlikely, mind you.

But really you absolutely need to have reactionless (which is to say impossible) drives to do 1 g for any astronomically significant period of time.

However, we should be able to build a solar powered soft launch plasma packet accelerator and a separate ship for it to push. Essentially you fire the fuel out of a cannon at a shield / sail so the ship doesn't have to carry its fuel. The resources required are considerable and the cannon facility itself has its limits. The cannon magnetically accelerates ionized matter like the stuff in a plasma TV rather than the hot death ray plasma of science fiction.

It's not 1G for a year by any means but it's probably the best bet at present. You probably do gravitic (use the target star's gravity to slow you down) or high atmosphere braking using the same sheild at the other end.