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Inspired by the movie, "The Core".
Can we really travel through earth's core? I will provide 2 sub questions:
Is there any substance that can resist the heat of earth's core?
Between the crust and mantle, and the mantle and outer core is there any "wall" between them? And how hard is the wall (can we go through it)?
As Chris Mueller said, in short: it isn't, or at least highly infeasible. Projects to drill into the mantle, such as the Kola Superdeep Borehole, have all failed because drilling equipment can't withstand the heat at only ~15km deep. Even if we were to come up with some sort of cooling system that's able to cool to 6400km or 12800km deep (depending on whether you would drill from one side only or from both sides at the same time), pressure is the second barrier that holds us from traveling through the earth's core.
According to Lide (2006) the pressure in the inner core is 330 to 360 GPa, at which iron becomes a solid even at the high temperatures in the core. If you could drill as far as the core you would have to build a device that's able to withstand that pressure, because if you can't, the material surrounding your well would immediately become liquid and fill the hole, if not shoot up your well towards the surface.
There are no physical walls between the layers of the Earth, only transition zones where temperature and pressure combinations lead to different behaviour of the materials. An example is the Mohorovičić discontinuity, or Moho, which is the boundary between crust and mantle, below which temperatures are high enough and at the same time the pressure is low enough so that rock becomes either liquid or at least a "flowing" solid. Similarly, at the boundary between the inner and outer core the pressure is so high that even at those temperatures the iron becomes a solid.
Lide, D.R., ed. (2006-2007). CRC Handbook of Chemistry and Physics (87th ed.). pp. j14–13.
Short answer: No.
Long answer: Our deepest drills failed around 12km down when the drill bits were having to cope with temperatures hot enough to melt the drills.
12km down is only a tiny distance into the earth. The average distance to the center is over 6300km. So didn't even get to half of a percent of the way to the center.
To do so, we'd have to have:
none of which are currently possible.
There is one vaguely plausible method that has been proposed: blow open a crack in the Earth's crust using a hydrogen bomb and then fill the crack with molten iron. The iron then sinks to the core, and a small probe can sit on top of it and ride it all the way down. There's no way to get it back again, of course, but it could transmit its data back to the surface using seismic waves. (We have detectors sensitive enough to pick them up.)
Although it's a pretty crazy idea, it was published in Nature magazine by a respected geophysicist, so there's a certain plausibility to it, although I don't think anyone intends to carry the plan out in practice. (Original article: Mission to Earth's core — a modest proposal, unfortunately paywalled. Googling the title will give a number of press articles about the idea.)
Nuclear reactions would be required to create a shaft. High melting point solids under the extreme pressures would be required to reinforce it. If gamma rays can tunnel, the theory is not implausible but requires an engineering feet to put physics into practice. This should be the focus of exploration besides space exploration.
