the game (perhaps on PS4), can you travel to stars, fly very near them, or are they off limits ?

On PS4.
As far as I can tell, every star is a planet or moon. And there are 18,446,744,073,709,551,616 of them.
You fly to them, continue to fly through the atmosphere, over the surface until you find a spot where you want to land...and then land.

You would need to upgrade your ship to leave your actual system. Otherwise it would take ages to fly to a nearby solar system.

Tom

Thats not what was asked.
Planets or moons orbit stars. Of course you can land on planets and moons.
But what about the star itself. In a lot of games you can't even go there or just die if you get to close.

Dying if you get too close is exactly what I'd like to see in NMS. The radiation, and heat of course, will just be too much for the ship, or yourself at least, to take for so long. You'll essentially be cooked inside your suit / cockpit before the vital components of the ship suffer the same fate and fail, in case of the engines / reactor meltdown probably a bad thing. Though, most likely, you wouldn't live long enough to experience that for yourself.

*Addendum:
Yes, space is a vacuum, yes, vacuum doesn't care about the surface temperature of ~5.7k °C, is it? Of the sun, but the IR radiation is what transmits enough of that heat through space and to anything within space. The further away from the sun, the more spread out these rays are, and the more it needs to heat things up. Get too close, and even your suit is big enough to catch enough of the IR radiation to overheat and cook you, way before it's melting itself. Take a ship with more protection and better life support capabilities, and its size cancels out the benefits these things bring. Since it's catching more IR than smaller things.

Ah,sorry. My bad.
That I cannot answer yet, but once I am ready to leave my actual planet I will give it a try.
I sure hope I will get a warning before I turn my ship into a ball of fire :)

As stated above; I would hope you get a warning WAY before that point, or else you would have been cooked to death in the cockpit a long time ago already and probably couldn't do much about that warning anymore.

Okay, I used my Pulse Engine several times now, but it looks like I only got further away from the planets and not really closer to the sun.

I think the planets in one system are always quite close together, and their distance to the sun only affects the environment on the planet, but the sun itself cannot be reached.

Yes, a star itself would completely vaporize any ship that gets too close, long before it could reach the surface of a star--planetary science 101, eighth-grade level. The game, however, should allow you to do that if you choose, imo.