Is it possible to sail "close hauled" with a solar sailship? I have not been able to find a clear or decisive answer about this. On Earth, a ship sails close hauled when the ship is, to some degree or another, facing into the wind. This is made possible with sails that are rigged along the length of the ship rather than perpendicular to the length. Is a similar thing practically or at least theoretically possible with a solar sailing vessel, to sail towards a star in this manner? Why or why not?
 A: A sailboat can make headway against the wind because of the sum of force vectors due to the wind interacting with the sail and, due to the keel interacting with the water.  A sailboat without a keel can not make headway into the wind.
There is no "water" out there into which a solar sailer could dip its keel.

http://newt.phys.unsw.edu.au/~jw/sailing.html
http://web.mit.edu/2.972/www/reports/sail_boat/sail_boat.html
A: Yes, but not the way you think. It's not analogous to a sail boat. A solar craft has no keel to resist movement in one direction, and I doubt that solar wind is dense enough to cause a noticeable wing effect with a sail.
But stars have their own complexities you can exploit instead.
Every answer thus far considers a static model where the star sits in space and you try to approach it using headwind. In this simplified model it's hard to approach the star using headwind. In reality, the star is almost certainly moving perpendicular to your path to some degree. This suggests that the solar wind will approach your sail from some oblique angle. Other factors will also alter the direction of the solar wind, such as the stars magnetic field, and any gravity the solar wind was exposed to.
While you may not be able to head upwind, you can certainly head downwind at any angle you choose with an appropriate sail. Use the apparent wind direction and your sail to intercept the star where it will be in the future. As you accelerate the apparent wind direction may well improve, allowing you to plot a more direct intercept course.
In the end, this is more of an engineering problem than a physics problem. Be creative.
A: The question is about solar sails - I thought I would add this (too long for a comment) to clarify some confusion that is apparent in answers and comments.
Solar sailing uses photon momentum, not the solar wind.
The radiation force on a perfectly reflective sail of area $A$ is roughly
$$ F_{rad} = 2\times\frac{L A}{4\pi r^2 c}$$
where $r$ is the distance from the Sun, $L$ is the solar luminosity and we assume the light is normal to the sail.
For the solar wind, the force is
$$F_w = \frac{\dot{M} v_w A}{4\pi r^2},$$
where $\dot{M}$ is the mass loss rate (assumed isotropic), with a value of about $2\times 10^{-14}\ M_{\odot}$/year and $v_w$ is the solar wind speed of about 400 km/s.
Thus
$$\frac{F_{rad}}{F_w} = \frac{2L}{\dot{M}v_w c}$$
Numerically, for the Sun this ratio is $\simeq 5000$, so you wouldn't design a sailship that used the wind! (The wind would be more effective near red giants or possibly very young low-mass stars).
As solar radiation travels radially away from the Sun, it is not possible to exert a force on a solar sail radially inwards. It is however possible to angle a reflective solar sail so that the force is almost tangential to the Sun, since the force is directed at right angles to the mirror surface. Navigating towards a star would largely involve slowing down the sailship, and allowing gravity to do its thing (see picture from https://directory.eoportal.org/web/eoportal/satellite-missions/n/nanosail-d ).

A: Yes, sort of, sometimes. As others have indicated, it is not possible for a solar sail to produce a force in the direction of the Sun. This does not, however, mean a solar sail cannot take a spacecraft to the Sun. If your spacecraft is in solar orbit, you can tilt the sail "backwards", reducing the orbital energy and lowering the orbit. The potential energy of the spacecraft decreases, its kinetic energy increases (by a smaller amount), and the rest of the energy is given to the photons, making them bluer.
