Well, right now we are trying to figure that out. NASA's Parker Solar Probe Spacecraft is currently orbiting closer and closer to the Sun. It recently completed its 3rd pass close to the Sun at 15 million miles away from its surface. The plan is for it to get close and closer until it swoops by less than 5 million miles from the surface. 5 million miles might seem like a lot, but it is much closer than we are - Earth is about 93 million miles from the Sun, and the closest planet to the Sun, Mercury, is about 36 million miles away. Parker is specially designed to handle the immense radiation form the Sun and still return useful data from this region of the Solar System we have never visited before. It is very tricky, though, and the Parker team is closely monitoring the spacecraft to keep it safe and healthy.
Here are some Parker Solar Probe links:
parkersolarprobe.jhuapl.edu/
https://blogs.nasa.gov/parkersolarprobe/

Another interesting measure of how close something can get to the sun and survive is Comet Lovejoy. In 2011 this comet passed about 75,000 miles from the solar surface and came back out again. It was amazing - most comets that get that close evaporate.
https://science.nasa.gov/.../16dec_cometlovejoy/

Of course I would not want to be a human being on Comet Lovejoy or even Parker Solar Probe. Designing something that could get a human safely that close to the Sun would be an entirely different kettle of fish!

Terry