A spacecraft that would allow humans to reach every corner of the Universe?

Currently is just a dream, even if what gives us hope today is the idea of David Burns, a NASA engineer who has proposed his concept of helical engine: a space engine that could, at least theoretically, accelerate reaching speeds very close to that of light (299.792.458 m/s), without using propellant!

According to the researcher report published on the Nasa Technical Reports Server, the helical motor could exploits the effects of mass alteration that occur when light speeds are reached.
Or, rather, the engine exploits Newton's principle of action and reaction at a relativistic regime, i.e. in which objects are accelerated (and decelerated) at speeds close to those of light, as happens in particle accelerators.

The engine accelerates ions confined in a loop to moderate relativistic speeds, and then varies their velocity to make slight changes to their mass. The engine then moves ions back and forth along the direction of travel to produce thrust. This in-space engine could be used for long-term satellite station-keeping without refueling. It could also propel spacecraft across interstellar distances, reaching close to the speed of light. The engine has no moving parts other than ions traveling in a vacuum line, trapped inside electric and magnetic fields.

NASA Technical Reports Server (NTRS) - Helical Engine

Unfortunately, this is only theory

According to this article on the NewScientist, the engine should be really huge and energy hungry:

It would also need to be big – some 200 metres long and 12 metres in diameter – and powerful, requiring 165 megawatts of power to generate just 1 newton of thrust, which is about the same force you use to type on a keyboard. For that reason, the engine would only be able to reach meaningful speeds in the frictionless environment of space. 

The same article tryis to explain how the helical engine could works:

…picture a box on a frictionless surface. Inside that box is a rod, along which a ring can slide. If a spring inside the box gives the ring a push, the ring will slide along the rod one way while the box will recoil in the other. When the ring reaches the end of the box, it will bounce backwards, and the box’s recoil direction will switch too. This is action-reaction – also known as Newton’s third law of motion – and in normal circumstances, it restricts the box to wiggling back and forth

But what if the ring’s mass is much greater when it slides in one direction than the other? Then it would give the box a greater kick at one end than the other. Action would exceed reaction and the box would accelerate forwards.

This mass changing isn’t prohibited by physics. Einstein’s theory of special relativity says that objects gain mass as they are driven towards the speed of light, an effect that must be accounted for in particle accelerators. In fact, a simplistic implementation of Burns’s concept would be to replace the ring with a circular particle accelerator, in which ions are swiftly accelerated to relativistic speed during one stroke, and decelerated during the other.

NASA engineers 'helical engine' may violate the laws of physics | New Scientist

So, is it possible?

The engine would only be able to reach meaningful speeds in the frictionless environment of space.

“The engine itself would be able to get to 99 per cent the speed of light if you had enough time and power”

David Burns