Going fast in a straight line with conventional rockets is achievable, just not very timely.
Bending spacetime enough to make a difference — even just to the time taken between here and Alpha Centauri — is beyond any known mechanism humans could build, even in principle, using the total resources of our entire solar system.
(Using unknown mechanisms: perhaps, but they’re unknown)
> Going fast in a straight line with conventional rockets is achievable, just not very timely.
A constant-acceleration spacecraft could reach the opposite side of the galaxy in 24 years ship time. (That'd be over 100,000 years of Earth time, however.)
Conventional chemical propulsion don't have high enough impulse to do that. Possibly some kind of nuclear or matter-antimatter propulsion could?
> When it comes to [high thrust, high specific impulse] propulsion systems we might actually be able to build in the near future, the list includes Orion drives, Zubrin's nuclear salt water rocket, and maybe Medusa.
There have been many studied variations of Orion. Atomic Rockets cites hypothetical Isp between 3,000 and 12,000 seconds. Vehicles with delta-V of up to 100,000 m/s have been proposed. These could definitely be built, and would definitely work, but would require constructing, co-locating, launching, and detonating thousands of 5- to 15-kiloton nuclear explosives.
Medusa is basically Orion crossed with the image of a sailor blowing into their own sails. A huge parachute is deployed in front of the spaceship, and the nuclear explosives are detonated between it and the bow of the spaceship. The parachute can capture more of the explosive output than Orion's pusher plate can, it weighs less (because all of its members are in tension), and it can use its rigging's elastic properties to dampen the intermittent thrust, as opposed to Orion's (heavier) hydraulic dampers. The proposal study for Medusa (which assumed thirty 25-kg explosives) came up with an Isp of 106,220 seconds, and a delta-V of 4775 m/s.
The Nuclear Salt Water Rocket assumes plausible (but completely untested, so maybe impossible) continuous nuclear fission in a stream of uranium-salt-y water being sprayed into the combustion chamber. Assuming that this can be made to not blow itself up, 20%-enriched salt nets a theoretical Isp of ~7,000 seconds, and 90%-(weapons-grade) enriched fuel, with somewhat more optimistic efficiency assumptions, has a theoretical Isp of ~400,000 seconds and a delta-V of 10,000,000 m/s.
There are also lots of other currently-less-plausible proposals for high-power high-efficiency rockets. (Many of them assume things like efficient proton-proton fusion, which might be possible, but we have no idea how to do it.) Atomic Rockets is a great website for reading about these.
My takeaway is that it's almost certainly possible to build spaceships that can travel at >0.01c, but it would take a great deal of resources organized by a society that is more trusting and responsible than we are now. After all, any vehicle with that kind of power is also an equally powerful weapon.
Everything would run out of fuel over that distance if you were constantly accelerating ahead [0], but you don’t need to constantly accelerate in space.
[0] spinning is accelerating, and you can do that forever, but it won’t help you get anywhere
Bending spacetime enough to make a difference — even just to the time taken between here and Alpha Centauri — is beyond any known mechanism humans could build, even in principle, using the total resources of our entire solar system.
(Using unknown mechanisms: perhaps, but they’re unknown)