At the speed of light and assuming a lifespan of 60 years, the answer to how far we can travel in a lifetime depends on the motions we include. The speed of the solar system around the Galactic center is about 230 kilometers per second. If we only include this motion, then we are traveling 726 billion kilometers a year, or 479 billion kilometers in total. If we want to “get out more” in stellar terms, we can consider inventing an antimatter engine that can take us up to 99.9 percent of the speed of light (and slow down again).
If we wanted to travel with a constant acceleration of 1G and then a deceleration to the very edge of the observable Universe, we would cover some 62 billion kilometers in 66 Earth years. If we accelerate at that rate for years, we can travel billions of light years over a human lifetime. However, individual human beings don't live billions of years, so they couldn't travel billions of light years at the speed of light. The longest distance that could be traveled while traveling at the speed of light during an entire human life would be only 71 light years.
It has nothing to do with “time travel”, but only with the reference point from which the measurements are made. When we approach at the speed of light and look out the window, we see that our destination is much closer.
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