Traveling "into the future" is possible via
Time Dilation (which, BTW, is an actual occurrence that GPS satellites have to account for in their calculations up in space). If you put somebody in a spaceship traveling art 95% the speed of light for three weeks, when they arrive they'll find out that even though they have aged three weeks, Earth has had 10.5 weeks pass on it.
Get closer to the speed of light and the multiplication gets even larger, to the point that simulation of
forward time travel (step into a space machine for 5 seconds, step out and the Earth is five years later) from a human's point of view is possible... but it might as well be a form of cryogenic freezing at that point, since the machine and the humans within it would actually still exist and need to be undisturbed through those five years (from the perspective of people outside the ship's movement).
Of course the energy/propulsion required to power a ship with reasonable mass near lightspeed is effing huge, so making
large differences in elapsed time for something as heavy as a group of people isn't within our reach right now.
On the other hand... traveling into the past, in my opinion, is not possible. We often talk about paradoxes "coming up" and "existing" in logical arguments because logical arguments are just human ideas. However in the real world, the laws of causality get in the way, and paradoxes are impossible by nature (which is why them coming up in a logical discussion is a point to be noted, since a paradox is a wall you need to stop at and back up from). A paradox cannot actually exist (even though humans and fantasy stories humans create toy around with the idea so often), so anything that would cause a paradox will not happen.