Hi Lea,
That depends on the technology you are using. I assume also you want to come back!
Right now NASA is working on a mission called Solar Probe which would swing in as close as 13 million km (8.2 million miles) from the Sun's surface 19 times and as close as 6 million km (3.7 million miles) 3 times.
At one point NASA was looking into a version of the spacecraft that would have gone 2 million km (1.3 million miles) from the Sun's surface, but only once or twice. It was decided that more times further out would be better.
In either case the spacecraft would need lots of special shielding to avoid frying the scientific instruments.
cheers,
Terry