Physicist: Hells no.
There are a lot of conclusions you can draw from the theory of special relativity that rule out FTL (faster than light) travel. From your own point of view you may as well be sitting still (try it), but from the perspective of someone watching you approach the speed of light all kinds of thing get messed up. But all of those bizarre effects can be boiled down to one solid principle, the “Einstein Equivalence Principle”, so I’ll just use that. The EEP states that the laws of physics operate exactly the same no matter where you are or how fast you’re moving. This includes the physics behind electro-magnetism, which dictates the speed of light. So a direct consequence is that the speed of light is the same, no matter where you are or how fast you’re moving.
The best way to tell if something is going faster than something else is to race them and see which wins. So imagine a photon whips by you at the speed of light (hence the name), and you start running (using a rocket, or whatever) to catch up. A minute later you’re going much faster, but the photon is still moving at the speed of light compared to you. In fact, no matter how fast you get going, light will always pass you at the speed of light. So it’s not like the race is even close. As far as the photon is concerned, you may as well not be moving.
So if what you think of as “speed” (miles per hour) gets messed up, then what you mean by “distance” and “time” (miles and hours) must get messed up as well. This is the (unfortunately correct) conclusion that Einstein drew. So, unlike breaking the sound barrier, which was an engineering challenge, but known to be possible, breaking the light barrier is known to be impossible, by everything we know about modern physics.
The most powerful fuel source that humankind is ever likely to come across is hydrogen fusion, which can get a ship up to about 11% of light speed. A “Bussard ram scoop” could get you going arbitrarily fast by collecting fuel en route, but there’s very little of it to be had (it wouldn’t be called “space” if it was full), so your acceleration would be very slow. An anti-matter rocket could also get a ship going arbitrarily fast, but there are problems with anti matter: 1) where do you get it?, 2) you can’t ever touch it (it’s the most unstable, dangerous stuff possible), 3) to get up to 90% of light speed half of your ship would have to be matter/anti-matter fuel.