Is it possible to drill a hole through the earth and come out the other end?

There are a lot of misconceptions that have to be cleared up.

1. The inner core of the earth is not a liquid, it is a solid (hot enough to be liquid iron, but under sufficient pressure to solidify). The inner core is not a sphere, but ragged.

2. Gravitational acceleration is based on the mass of the sphere under you (ignore the mass above you). That is, if you are half-way to the center of the earth, gravitational acceleration is only from the half-earth diameter sphere below you (and the mass above you cancels its gravity with the remaining earth mass).

3. If you fall through the earth and come out at the same elevation, you could step out onto the surface of the earth, or you could let yourself fall back through again and end up where you first started. This is because gravitational fields are conservative (this is easier to understand if you consider that the potential energy on one side of the earth is exactly the same as the potential energy on the other side). Except for friction, nothing would keep you from bouncing back and forth from one side of the earth to the other. It is possible to depressurize the tunnel, taking out the air that makes friction, and seal yourself in a pod with air, and travel endlessly from one end of the earth to the other.

4. The heat in the core of the earth is intense, so the tunnel would have to be insulated (or somehow cooled). Perhaps wires could be run through the tunnel, cooling by the Peltier Effect?

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It would be a very cheap way to travel (once the expensive tunnel is constructed).

No, because the gravitational force at the center of the Earth will pull you right back. It's possible theoretically to drill a hole from both sides of the Earth and connect at the center. To get from one end to the other, however, is not so simple. You'd gain kinetic energy from when you jump in to when you get to the center. At this point, you'll need to overcome the gravitational attraction of the Earth, which at this point, is infinite due to the equation:

F = Gm1m2/r^2, and since you're at the center, r =0, so theoretically, you'll be stuck there forever.

Since the core of the Earth is molten, as in liquid, it would be like trying to drill a hole in water in a glass. Won't work.

The earths core is too hot...you will burn waaay before u even get halfway through.

Till now no one could do but may

be one day some one will try to do .

I did this last night

it's impossible & can compare it to this. climbing down someones throat & coming out of their anus.

http://hyperphysics.phy-astr.gsu.edu/hba%E2%80%A6

Sure why not