The brightness and warmth of the sun hold a certain magnetism. Our eyes are drawn upwards to stare and soak in that warmth, and maybe there's even a draw in the challenge, or danger, of staring too long at the sun, until we see spots, in defiance of mothers everywhere.
Danny Boyle's Sunshine put a sci-fi twist on that instinctive human desire to stare at the sun, giving the ship's doctor Searle a dangerous solar obsession. In the film, the observation room of the Icarus spaceship, which is heading ever closer to the sun, can dim the star's brightness to survivable levels, but Searle lowers the dimmer more and more as his obsession grows, exposing himself to greater amounts of solar radiation.
On Earth, staring at the sun can't offer the same intense experience--we're much, much further away from the sun than the Icarus was in Sunshine--but there are ways to stare up at the sky without seeing spots and getting watery eyes after a few seconds. The Atlantic recently wrote about the experience of staring skyward from Aspen, nearly 8000 feet above sea level. Solar viewing glasses from the Charlie Bates Solar Astronomy Project are one cheap and simple solution. They only cost about $1 per pair, and make the sun safe to stare at:
"All of our solar viewing materials are optical density 5 or greater and are "CE" certified which meets the transmission requirements of scale 12-16 of EN 169/1992. Lenses are made of our exclusive scratch resistant optical density 5, "Black Polymer" material. Eclipse Shades filter out 100% of harmful ultra-violet, 100% of harmful infrared, and 99.999% of intense visible light. These premium filters create a sharper ORANGE colored image of the sun."
The second option, though, is far more compelling: Using a solar telescope to see sunspots, flares, and other activity on the surface of the sun. Solar telescopes like this one aren't cheap, of course, but filters are pretty affordable, and you can always build one yourself. There are all kinds of tutorials online.