That mostly depends on whether or not matter with negative mass can exist in this universe. If it can, then the Alcubierre drive is possible (once the insane power requirements are resolved). If not, then he best that we know is definitely possible is around 0.1C. That gives us a transit time of 10 years per light-year, or around 40 years to the Centauri system. That's with Project Orion (a rocket-like vessel powered by literal fusion explosions).
So unless we can significantly increase the fraction of C it would probably be literally a once-in-a-lifetime thing, if that, for most people.

There are some theoretical possibilities, like replacing the fusion explosions with antimatter explosions (or even some sort of actual antimatter rocket) or the Alcubierre drive. Both of them require currently-unattainable quantities of special materials.

Any sort of antimatter-powered vessel capable of carrying people is going to have a mass measured in thousands of kg (probably many of them) and will require facilities for either storage or in-situ production of similar mass of antimatter for a given trip. Somewhere around 0.2C is where the payload:fuel mass ratio hits 1:1 (assuming in-situ production isn't an option, and of course making assumptions about the maximum efficiency of the engine).
We can currently only make antimatter as individual subatomic particles in particle accelerators, and our current facilities aren't designed for that. With current tech, a facility optimized for it could supposedly make about 20 grams/year. That means 1 tonne (1000 kg) would be 50,000 facility-years of antimatter production with today's technology. I think it is highly likely that this will change in the future and we will actually be able to produce enough to use as an exotic power source for special applications. I suspect that the energy cost to produce it will always remain prohibitive compared to other manufactured fuels, but I'm also sure that at some point our energy production capacity will (eventually) trivialize that by brute force.

As for the Alcubierre drive (which would theoretically allow travel faster than C), it's dependent on a hypothetical type of exotic matter that has negative mass. It isn't used as a fuel, but to generate the negative gravity well on one side of the "warp" field. We literally don't know if such matter can even exist in this universe. It's mathematically predicted by some symmetries, but that doesn't necessarily mean it actually exists. That could simply mean that something else is missing from our models. At any rate, even with the Alcubierre drive generating the field, there still needs to be some kind of energy source to empower the drive, and it takes a lot of power. We're talking planetary masses' worth of fuel even for hypothetical antimatter reactors (and I mean gas giants).