“How far away from Earth do we have to go to not see it with our own eyes?”
To answer this question, you have to take into account how brightly Earth reflects sunlight. And the sun itself is another important factor. As seen from any great distance, Earth appears right next to the sun – and, from a great distance, the glare of our local star would make Earth difficult or impossible to see.
So imagine blasting off and being about 300 kilometers – about 200 miles – above Earth’s surface. That’s the height at which the International Space Station (ISS) orbits. The surface of the Earth looms large in the window of ISS. You can clearly see major landforms, and the lights of cities.
As you pass the moon – about 380,000 kilometers away – or a quarter million miles – Earth looks like a bright ball in space – not very different from the way the moon looks to us.
Speeding outward, you pass the orbits of the planets Mars, Jupiter, Saturn, Uranus and Neptune. From all of these worlds, Earth looks like a star – which gets fainter as you get farther away.
The light from Earth finally becomes too faint to see with the eye alone at around 14 billion kilometers – about 9 billion miles – from home – around the outer limit of our solar system – nowhere near as far as even the next-nearest star.
By the way, the Voyager 1 spacecraft is now farther away than that. In February 2012, it was about 18 billion kilometers (11 billion miles) from our sun.
If an astronaut/alien had a telescope, he or she could definitely see Earth further away than that.
Bottom line: How far away in space can you view Earth with the eye alone? About as far away as the outer reaches of our own solar system at about 14 billion kilometers, or 11 billions miles.