No, but we probably could with enough fuel. The bigger issue is space dust. Hitting it at relativistic speeds would be like flying into a cosmic shotgun for the entire journey.

According to this paper, you can travel around 0.5 c before the radiation caused by colliding with interstellar hydrogen kills you: http://www.scirp.org/journal/PaperInformation.aspx?paperID=2...

"Going slow to avoid severe H irradiation sets an upper speed limit of v ~ 0.5 c. This velocity only gives a time dilation factor of about 15%, which would not substantially assist galaxy-scale voyages. Diffuse interstellar H atoms are the ultimate cosmic space mines and represent a formidable obstacle to interstellar travel."

Couldn't you put some kind of "snowplow" on the front so it pushes the oncoming detritus away to the sides?

A reddit thread mentions that idea: https://www.reddit.com/r/science/comments/12nls5/speed_kills...

Could that cause accidental fusion?

Assuming this [1] is reasonably accurate, very much yes. It proposes fusion occurring at speeds as "slow" as 0.9c.

[1] https://what-if.xkcd.com/1/

A large mass of water ice would make a decent ablative shield.

If I understand our capabilities correctly we can't even escape the solar system without using gravity assist boosts from other planets, let alone sustaining thrust for a long period.

http://solarsystem.nasa.gov/basics/chapter4-1

"Capabilities" seems a little ambiguous to me. Is it what we have done or what we could do? Using those boosts is a way to achieve the same results, only cheaper. Yes, it's slower than building a big ass rocket, but it seems people that sign the checks are in no hurry.

Also if I'm not mistaken, Voyager probes were not even intended to live so long or leave the system.

If there were a reason to send a probe to another star as soon as possible and no matter the cost, there would be a way, maybe not fast, maybe very expensive, maybe something weird, but it would be done.

I've read about laser systems for sending unmanned crafts at high speed: http://www.space.com/29950-lasers-power-tiny-interstellar-sp...

I think we do if you look at it in terms of acceleration. As I understand it, In space there is almost 0 friction so "fast" is more about acceleration than "top speed". In this view, 1g isn't that fast.

About as "fast" as a helicopter at a stable altitude, or to fit with the no-atmosphere bit, a stationary jetpack or lunar lander (on Earth).

Doesn't energy required to accelerate keep exponentially increasing, though? ke = mv^2

Energy is relative to a frame of reference, and IIUC 1g acceleration of the space vessel is in The vessel's frame of reference, not earth's.

No. If I'm not mistaken we're in the order of magnitude of 0.0001c.