Locomotive can harvest energy and matter from Sun and then use it to orbit around Sun, slowly gaining speed, until it will no longer will be able to compensate сentrifugal force. Then it can reach high orbit passively, pickup pilots on high orbit, and be ready to flight, with full tanks. Or it can dump fuel, and do another dive to Sun, for more fuel.
If it will survive Sun radiation at low orbit, then it may also survive interstellar travel at high % of C.
From that article the best case scenario is a billionth the acceleration you are looking for. And you can’t take a billion stars because then you have more mass...
Edit: the interesting part of discussing the 1g is probably less a realistic proposal and more how about how the concept of acceleration quickly runs up against our mental constructs of what acceleration and distance means. I don’t know the source, but it’s probably an enlightening to discuss what it means in the context of relativity.
Ok, it would be exciting to move the whole solar system to a nicer place in the galaxy. But how many years are we not going to use to evaluate the ethics and technical risks of the experiment? :)