Highly accelerate space travel is possible, but has many things that still need to be solve (beyond figuring out how to travel near speed of light, though plenty of theoretics out there that show it's possibility and mostly mimic the same issues).
(1) Energy - Need to have massive energy and ability to sustain or kick in occasionally that energy to keep the speeds. Yes when something is given a boost in space it will move very long distances will relatively little loss in speed...however there are many gravitational forces in effect that will accelerate that slow down over long distances. There is a reason why satellites and other space craft use the sun or moon to get a boost.
Plasma (or Ion) engines have existed since the 70's and there are pretty well refined to the point of getting a large space craft to Mars in about 40 days (vs 6+ months). However, Plasma engines need a large amount of energy to kick it off and gain those speeds. While that energy can be generated in a lab, there is still the need to be able to do it on a space craft.
Liquid, gas, or solid fuels can be used, however they are very bulky and heavy. Also need to get the craft, or parts of the craft into space, which has it's own issues with the weight and bulk. So, in the last decade the research focus has gone towards nuclear. Not a reactor, per se, but more of a battery. While nuclear will be lighter and sustain energy output much longer than other known fuel options, it is still too bulky and would eat up too much critical real-estate on the spacecraft. I think this issue will be solved in the next 5-8 years given the money going into it.
(2) Protection - Lots of nasty stuff that will degenerate the body very quickly once it leaves the majority of Earth's atmosphere and gravity. Not too much advanced research here since #1 still needs a ton of further research. But it will need to be resolved regardless of the distance and time of travel.
(3) Food and other resources - Still need to eat, drink, and handle body waste on those long trips. The ISS has been a live test that has shown major strides in this area. Everything from hydroponic systems for food that get the water and nutrients from body waste...to filtering out urine to get clean water to drink. However, nothing that shows a highly long sustainability without addition resources to assist. These processes also eat up a ton of space.