The only real possibility of us not exploring our galaxy is self-destruction. There are enough nuclear weapons lying around to destroy all life on Earth a few tens of times over, and there are groups who'd be willing to do it for all sorts of reasons. Hell, there even are christians who feel it is their duty to start a nuclear war to hurry yehoshua's second arrival on Earth.
That being said, if our species survives the next decades, I'm very confident that in the next centuries, we'll have explored other star systems, in the vicinity of ours. If I'm lucky, I might even travel to Jupiter and back while I am still alive.

I don't think there's any natural barrier between us and space exploring. Humans have already escaped Earth's atmosphere for a lot of times, the discussion here isn't about breaking some natural laws, but about perfecting ways which
we already have of using the natural laws to fly.