The closest stars in the universe aren't separated by light-years or even miles. The distance between them is exactly zero—they are physically touching.
In our own solar neighborhood, stars are kept far apart. Proxima Centauri is the closest star to the Sun at 4.24 light-years away. To put that void in perspective, if the Sun were the size of a ping-pong ball in New York City, Proxima Centauri would be a slightly smaller ping-pong ball in Chicago.
The average gap between stars shrinks in the core of a globular cluster or near the galactic center. In these densely packed regions, stellar density is millions of times higher than in our spiral arm. Stars can be separated by just a few light-weeks or light-days. A planet orbiting a star in one of these clusters would have a night sky filled with thousands of stars bright enough to cast shadows.
The ultimate extreme, however, is found in binary systems. More than half of all stars exist in multi-star systems, orbiting a common center of mass. Over millions of years, some of these stellar pairs spiral inward until they become a "contact binary."
The system VFTS 352, located in the Tarantula Nebula, is one of the most extreme known examples of a contact binary. Two exceptionally hot stars orbit each other in slightly less than 24 hours. They are so close that their outer atmospheres overlap, creating a single shared envelope of stellar plasma that bridges the gap between their cores. Because they are similar in size, neither star is cannibalizing the other; instead, they are mutually mixing their internal material. Systems like this are highly unstable and represent a brief phase in stellar evolution. They will eventually either merge into a single rapidly spinning giant or detonate as a pair of supernovas.