If you wedged the Solar System closely between two stars, the resulting gravitational tug-of-war would tear the planets out of their orbits and fling Earth into deep space.
In a purely geometric sense across interstellar space, the Solar System already sits between every pair of stars on opposite sides of the sky—draw a straight line from Alpha Centauri through the Sun, and it points toward the constellation Cassiopeia. At close range, however, orbital mechanics allows planets in two-star systems to survive in only two stable layouts. In a wide binary such as Alpha Centauri, where the two main stars orbit each other at a distance varying from 11 to 36 astronomical units, planets must hug one star closely so the second star does not rip their orbits apart. In a tight binary such as Kepler-16, the two stars orbit each other so closely that planets circle both of them from the outside, treating the pair as a single gravitational center.
Even at the outer edge of the Solar System, where the Oort Cloud of icy comets extends up to 100,000 astronomical units (roughly 1.5 light-years) from the Sun, no second star lurks in the dark. In 1984, researchers proposed the Nemesis hypothesis: a dim red dwarf or brown dwarf orbiting 1.5 light-years away that periodically disturbed Oort Cloud comets and sent them hurtling toward the inner planets. Between 2009 and 2011, NASA's Wide-field Infrared Survey Explorer (WISE) scanned the entire sky in infrared light sensitive enough to spot a cool, Jupiter-sized brown dwarf out to 10 light-years. It found no companion star.
While the Sun travels solo, the galaxy's stars are constantly moving, and passing stars occasionally thread the outer reaches of the Solar System. About 70,000 years ago, a dim binary pair known as Scholz's Star passed through the outer Oort Cloud at a distance of roughly 0.8 light-years. In about 1.3 million years, the orange dwarf Gliese 710 will pass even closer—within about 0.16 light-years, or 10,000 astronomical units—shining in Earth's night sky three times brighter than Sirius while briefly placing the outer Oort Cloud comets directly between the Sun and a visiting star.