How far could be an object from the Sun and still be under the influence of its gravitational field?
The Sun's gravity extends infinitely, but eventually solar objects would be unstable due to the influence of other stars. The minor planet "Sedna" has an orbit which takes it nearly 1000 AU (0.016 light years) from the sun at its furthest point (but now it is a lot closer)
It is also thought that billions of comets must orbit in the outer part of the solar system, out to 50000AU, or 0.8 light years, (or possibly further) forming the Oort Cloud. However, at such distances, they could not be directly observed. This marks the greatest distance at which orbiting solar system bodies can be found.
And for example, is there any limit of mass and distance? For example, jupiter can't orbit at 1000AU or it will fly away, or Saturn cant orbit at 2000AU or it fly away, is there any mass-distance orbit relation?
@AlbertoMartínez No, the mass of a planet has no influence on its orbit. The only thing you need to create a stable orbit is the right velocity. If you want you could place jupiter 1km aboves sun surfaces if you make sure jupiter has the right velocity
The word "infinitely" doesn't really have meaning here. Even gravitational waves propagate, so despite the simple 1/R^2 model, one can claim that the Sun's field reaches no farther than the speed of light times the age of the Sun. (yes, I know that's inaccurate since the sun didn't pop into existence)