This is how time travel is possible:
I always liked the theory postulated in the Red Dwarf episode "Backwards" where the beloved, goofy British space travelers from the hapless ship Red Dwarf travel to a different dimension quite by accident and arrive on an Earth where everybody and everything is moving backwards. People even talk backwards and signs are even written backwards. They theorize that, in this dimension, the universe had expanded all it was going to expand and was now shrinking, and hence since time and matter are related, the matter traveling backwards caused time to go backwards as well. This leads to the conclusion that the universe is forever in a cycle of expanding and contracting and that time is moving forwards and backwards, respectively, and right now we are just in one of the cycles.
Now how does this relate to time travel? Simple. It has always been a theory that travel to the past is impossible because, first off, it would create paradoxes, and second, you can't go anyplace that doesn't exist anymore, i.e. past events are over with and there's no way to revisit them. But time travel to the future is possible. However, in this expanding and contracting universe model, you can travel to the past by traveling to the future (which is possible by some theories of quantum physics); just go far enough into the future that you go to the end of the cycle you're in now, pass the backwards phase, past where there is another Big Bang, and then stop at whatever time period you wish. And to get back to where you came from, travel to the future some more. And there will be no paradoxes because if you change something it will be no more damaging that crap you're doing right now because it is the future and it won't screw up anything in your past or create a paradox.
Now all you need to do is build a time travel machine. I suggest using the very plausible method described in the book Time Line (for God's sake don't use the movie for reference) by Michael Crichton wherein via quantum foam the travelers are "faxed" through time using matter transportation technology. One of the main things needed for that is a quantum computer, a machine powerful and fast enough to crunch a human being into data (every cell, every molecule, every atom) and spit it out.
Or you could use a DeLorean.