In theory, yes. In practice, your journey through the planet might be hampered by the planet’s molten core.
There’s also the matter of finding a spot for all you’re digging through to make your tunnel. Let’s not rule out the possibility of mole people, dinosaurs and graboids living down there, either. Keep in mind, too, that you would need to build some sort of DIY digging contraption. Even the biggest and best digging operations in the world haven’t breached the Earth’s crust yet and to even get close, they had to start digging from the ocean floor. I’m going to guess you’re starting from your backyard, which means you don’t have a chance.
But we can dream, can’t we? And who am I to step on your dreams?
Let’s have a little fun and speculate, then, about what physicists say a trip through the planet might be like. To make your journey a little easier, we’ll assume certain ideal conditions:
1. You did your homework and know that if you dig a straight hole down in the United States, you’ll come out the other side not in China, but in the Indian Ocean. To avoid this very wet ending and get to China, you started digging in Argentina. Good for you.
2. You managed to actually dig a tunnel with your fancy homemade digging machine and found a place for all the rubble. You’re now standing there, peering into the hole and ready to jump in, passing Argentinians eyeing you warily.
3. The Earth’s core is not molten, so your digging machine did not melt and neither will you.
4. The Earth has the same density throughout. At the center, you have approximately equal amounts of mass on all sides of you, which cancel each other out and result in no net gravitational force acting on you.
5. The Earth isn’t rotating, which made it easier to dig your hole and will keep you from bouncing around in your tunnel and getting all bruised up.
6. There’s no friction, no air resistance, and no mole men.
update: some helpful readers have pointed out other conditions that I neglected in the original post...
7. Either Bernoulli’s principle doesn’t apply or you're wearing some sort of breathing apparatus and oxygen tank, that way your high travel speed won't affect your ability to breath.
8. The air pressure at your starting point, throughout your tunnel and at your end point is uniform, so you don't get squished into goo.
It’s a lot of concessions to make (and I'm sure we could even think of a few more), and we’re now on an Earth very much unlike the one we know and love. Whatever. It’s a small price to pay for the thrill ride you’re about to take.
So go on. Step into the hole. Or maybe dive in headfirst; you’ll have a better view. As you fall through your tunnel, gravity pulls you down towards the center, and you gain speed. As you get closer to the center, you’re closer to that balance of mass we assumed. Gravity doesn’t pull on you as much and while you’ll still gain speed, you won’t do it as fast. Once you hit the center of the Earth, you’ll be in zero gravity, but going at maximum speed (some 18,000 mph), you won’t even notice.
As you pass the center, gravity starts to work against you, pulling you back towards the core. You’ll start to decelerate at exactly the opposite rate that you accelerated during the first half of the trip. When you reach the opposite end of the tunnel (the trip would take you, appropriately, 42 minutes), you’ll come to a dead stop for an instant just as you pop out of the exit hole. Unless some considerate Chinese person happens to be near the hole and grabs you, all of Earth’s mass will pull you back towards the core and you’ll go back down (or up, as it were) the hole again.
If no one catches you at either end of the tunnel, you’ll spend the rest of your life oscillating back and forth, the human yo-yo at the center of the Earth.