If you want to fly a spaceship with human passengers as close to the Sun as possible, then what effects would the spaceship have to be designed to counteract in order to keep the passengers alive and how close to the Sun could you get before there would be no way to counteract the effects ?
Assuming you're in orbit around the Sun (presumably a highly elliptical orbit) you won't feel any force due to gravity. In principle you might feel tidal forces, but for an object the size of a spaceship these are negligable even if you graze the surface of the Sun.
The most obvious problems are the heat from the Sun and the radiation it emits. The radiation is a mixture of electromagnetic radiation and charged particles, both of which are not good for anything relying on it's DNA remaining intact.
It's difficult to do much about the heat because in space the only way you can cool is by radiation. What you'd probably do is surround your spaceship with a mirrored shell and keep a layer of vacuum between the shell and your ship. Even with very good mirroring the shell will heat up, but for a while at least it will keep the heat off your spaceship. The MESSENGER probe in orbit round Mercury uses a reflective shield, and contains internal refridgeration - I don't knw exactly how this works but presumably it uses a radiator on the side of the probe pointing away from the Sun.
There isn't a lot you can do about the radiation except surround your spaceship with a thick layer of lead, and that much lead would be difficult to put into space.
The Solar Probe Plus is planned to get within 4 million miles of the Sun's surface, and this will be the closest we've managed to get any spacecraft. However the SPP doesn't have any human passengers to worry about. I suspect radiation is the real problem for human passengers. Even for a hypothetical manned Mars mission the radiation dose the astronauts would receive is a worry, and the intensity of the radiation goes up as the inverse square of the distance.
If you get really close but have thick enough shielding to avoid radiation, and are travelling fast enough that you don't spend enough time close to heat up too much, then there still could be problem of Sun's magnetic field. If field is strong enough it could disrupt biological function.