I'm definitely handwaving away the difficulty, but I did explicitly speak to these concerns.
We really can skip the need for fuel, for example. Sails (with or without lasers) are a technology we have proven in the field. Lasers do lose collimation over distance, but you can reach relativistic speeds before then (I'd argue that .02c from 1wk of 1G is relativistic). That won't get you to the center of the galaxy (or solve the deceleration issue), but there are proposals being reviewed today to use lasers to send probes for a flyby of Proxima Centauri.
https://www.nasa.gov/general/swarming-proxima-centauri/
https://en.wikipedia.org/wiki/Breakthrough_Starshot
But we don't need science fiction tech for this to work, we just need impractical amounts of fuel. Starship only has enough fuel to last 10m? Just send 50k starships and you can burn for a year. Tyranny of the rocket equation requires additional fuel to push all that fuel? Just send another billion Starships or whatever. Going too fast and now the interstellar medium hits like high-energy cosmic rays? Just send more shielding and fuel. This assumes we can build and fuel billions of Starships, which is certainly infeasible, but I'm calling this an economics issue as we have these technologies today.
If we want to get really sci-fi, I'd point you towards stellar engines. The thought process here is that the Earth already provides radiation shielding, and the Sun already burns fuel to provide massive amounts of energy, so we might as well just make use of what we got! Add mirrors to concentrate the Sun's light in one direction, and our entire solar system becomes an interstellar spaceship. It might take millions or even billions of years, but the Sun has enough fuel to accelerate the whole system to about .27c.
https://en.wikipedia.org/wiki/Stellar_engine