I'm going to embarrass myself horribly here, but I've only seen the series, so I'm not a hundred percent clear on how much acceleration they're pulling in cruise[1]. If it's something like .3 g then they're fairly close[2] to the limits of something like a fusion pulse drive (tiny nodules of deuterium or whatever, ejected out the back of the ship, then lasered into fusing, then the reaction pushes on the whatever - magnetic field, pusher planet, etc). That would make the solar system something more like what the Atlantic Ocean was in the days of sail.

They'd need refuels every stop though. Like with early coal fired steamers back in the day.

I got a funny feeling that without much better genetic muckety muck, the hard limit will be the human organism itself. Someone will get to the stars = if we don't screw everything up - but the someone won't be human, or maybe nothing like human.

[1] I know in emergencies they pull mad g, but it doesn't seem like something they keep up for long. Would still burn through their deuterium in no time though.

[2] Like, within an order of magnitude or two.