I Agree with Mark Olson that having the nearest star appear no brighter than Venus in the sky is the main factor setting a minimum distance for the neighboring stars. And I think that half a light year away is not a very great distance improvement. That would allow the nearest star with the same absolute magnitude as the Sun to be 8.6 times as close as Alpha Centauri, or 11.62 percent as far as Alpha Centauri, which is very helpful, but still heart breakingly far for current space travel technology. It would be a much better improvement to make the nearest star one percent as far as Alpha Centauri, or 0.1 percent as far as Alpha Centauri, or 0.01 percent as far as Alpha Centauri.
If all the neighboring stars are very dim red dwarfs, or if it is permissible for them to look many times as bright as Alpha Centauri, they could be much closer than 11.62 percent as far as Alpha Centauri, and it would be much easier to reach them.
Note that the planets in our solar system have stable orbits despite being 4.37 light years or 1.339 parsecs, from the nearest star system, Alpha Centauri. So it is obvious that 4.37 light years or 1.339 parsecs is larger than the minimum separation between stars necessary for planets orbiting those stars to have stable orbits.
There are many double and multiple star systems in which two or more stars orbit each other. And if two stars in a binary system are far enough apart they can both have planets orbiting in stable orbits around them.
For example, Alpha Centauri C, or Proxima Centauri, is probably part of the same star system as Alpha Centauri A & B. The distance between Proxima and Alpha Centauri A & B is 12,947 plus or minus 260 Astronomical Units (AU), or 1.94 plus or minus 0.04 trillion kilometers. A planet was discovered orbiting Proxima Centauri in 2016. Planet Proxima Centauri b is estimated to have at least 1.3 times the mass of Earth and orbits at a distance of about 0.0485 AU with a period of 11.186 Earth days. Proxima Centauri b
orbits within the habitable zone of Proxima Centauri.
Since there are approximately 206,264.806 AU in a parsec, and Alpha Centauri is about 1.339 parsecs or 276,188.56 AU from the Sun, the distance from Proxima Centauri to Alpha Centauri A & B is about 0.0468773, or 4.68 percent, of the distance between Alpha Centauri and the Sun. And that distance is great enough for Proxima Centauri b to have a stable orbit around Proxima Centauri within the habitable zone of Proxima Centauri. (Proxima Centauri is a flare star so there is doubt whether Proxima Centauri b could be habitable)
Alpha Centauri A, or Rigil Kentaurus, and Alpha Centauri B, or Toliman, orbit each other at distances ranging from 11.2 AU (1.68 billion kilometers) to 35.6 AU (5.33 billion kilometers), or from about the distance of Saturn from the Sun to about the distance of Pluto from the Sun.
A planet of Alpha Centarui B, Alpha Centauri Bc, was announced in 2013. If it is real it orbits at a distance of about 0.10 AU and a year about 12 Earth days long. It is closer to Alpha Centauri B than the habitable zone and probably has lakes of molten lava.
This indicates that two stars can get as close as 11.2 AU without disrupting the orbits of their closest planets, though obviously planets orbiting at the distance of Saturn or farther out would have their orbits disrupted.
It has been calculated that the habitable zone of Alpha Centauri A would be about 1.25 AU out and the habitable zone of Alpha Centauri B would be about 0.7 AU (100 million Kilometers) out. It has also be calculated that planets in those habitable zones would have stable orbits, though none have been discovered yet.
If two stars can get as close as 11.2 AU to 35.6 AU without disrupting the orbits of their inner planets, then two stars can get within 0.0000405 to 0.0001288, or 0.00405 percent to 0.01288 percent, of the separation between the Sun and Alpha Centauri without disrupting the orbits of their inner planets.
The Wikipedia article Habitability of Binary Star Systems says:
In non circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.
This implies that if a planet's orbital distance is less than about one fifth of the closest approach of the other star, its orbital stability should be or might be guaranteed.
So if Earth was part of a binary star, and the nearest approach of the other star to the Sun was farther than about 5 AU, the orbit of the Earth would remain stable and almost unchanged.
However, these observations and calculations are for planets orbiting stars in binary systems where the two stars have elliptical orbits around their common center of gravity and thus travel at the correct orbital velocities.
Two stars separately orbiting the center of the galaxy that happen to pass close to each might have significantly greater relative velocities. And that might mean that they would have to pass at distances many times as great as 5 AU to avoid disrupting the orbits of planets in their habitable zones.
Anyway, such close passes would certainly disrupt the orbits of small bodies in the outer solar systems of the two stars which might result in multiple extinction events caused by asteroid and comet impacts on any habitable planets.