Do time machines really exist? No. No as such. Can they exist? The answer is an absolute...maybe. The physics is complicated, but there are some loopholes that might be big enough to fit a person through. I don't claim to understand all of it, but I'll take a stab at it.
Most people know what a black hole is. You take a star that is bigger than our sun, and wait for it run out of fuel. Depending on how big it is, and what type of star it is, it may collapse in on itself to the point where its gravity becomes so strong that not even light can escape (hence the black part of the black hole.) Now I have to talk a little about physics. Einstein's relativity theory (among other things) explains that the faster we go, the slower time becomes for us. It turns out that if we could get to the speed of light, time would stop. This explains the other part of black holes that many people have heard of, the event horizon. As an object falls into a black hole, it falls faster and faster. Eventually it is going so fast that reaches the speed of light and time stops for it. This is why the event horizon is the point of no return - if time has stopped, you can't come back. Some would also argue that relativity says you can't reach the speed of light in this universe, so by falling into a black hole you have reached the speed of light an fallen out of this universe.
Now comes the time machine part. If you spin a black hole fast enough (don't ask me how to spin a black hole...lets just say you found it that way), the event horizon sort of bulges out around the equator. It turns out it either separates into two event horizons, or gets thicker (I'm fuzzy on the details). Anyway, what happens is the volume of space-time between the two event horizons gets mixed up. It becomes either time-like space, or space-like time (or both). Traveling a distance in that volume is the same as traveling in time out in the rest of the universe, and traveling in time is the same as traveling in space. This means that you could theoretically pass through the outer event horizon, fly a little while and pop back out in a different time.
Neat huh? So now we need to either find a black hole that is spinning at near the speed of light, or make one. We will need a ship strong enough to survive the tidal forces that try to tear it apart as it approaches that black hole, and armored well enough to not be vaporized by the radiation given off by everything else that is being torn to sub-atomic particles as it falls in. Oh yes, don't forget the gravity shielding to stop the passengers on said ship from being turned into a read smear on the inside by the gravity...
Hey. I said maybe.
If you want to find out more, you can look up "Dragon's Egg" and "Starquake" by the late Dr. Robert L. Forward (science fiction, yes, but written by an astrophysicist). You can also look up "Kerr Space Warp." I think there might even have been a Scientific American article many years ago.