There is a system of one star and something like an asteroid belt around it. But the distance between asteroids is much less than usual, with some perhaps within nine kilometers of each other, and sometimes even less (there can be closer proximity, to several meters, and collisions are very frequent). Also, it is an inhabited place, so there's oxygen and something does not allow to fly it away. From inside this looks like millions of flying islands.
So how this can be possible?
Let's assume some area in space contained your "islands" - planetoids orbiting your sun. And somehow there was an atmosphere around them (we will get to that later). Atmosphere would have to orbit sun exactly same speed as your planetoids, otherwise they will experience drag, lose orbital speed, and fall to the sun. OK so far so good.
Any other space body passing area of your planetoids would slam into such atmosphere. Comets, meteors, you have it. Gravity from other big planets orbiting your sun (if any).
You would have hard time to keep such atmosphere where you want it - gravity of a planet keeps it around a planet in a "standard" case, but your "islands" do not have enough gravity to keep gases from dispersing in space.
So unless you use some complicated unstable construct like Niven, is "standard" universe answer would be: NO. Not without using a lot of magic or handwavium.
If we put many planets next to each other, going around the sun, so that it is possible to walk from planet to planet, in effect a torus shaped planet that would work. If the torus planet had huge lakes in it, then we would have actual islands orbiting the sun.
If the torus is like swiss cheese, then you could have islands and an irregular torus with missing parts, but these islands would be high mass in order to hold sufficient atmosphere for people.
So it would work, but the density of the islands would be very high, so it would be pretty warm in the torus. The density would be orders higher than an asteroid field. But then asteroids have low density.
