How big is the Solar System?
Short answer: The Solar System is a disc shape, 244 astronomical units wide (244 times the distance from the Sun to the Earth, or about 36.6 billion km).
Long answer: There is some disagreement over exactly where the edge of the Solar System is, so there are disagreements over its size. One common definition puts the boundary of the Solar System at a place called the heliopause, where the outward pressure of the Sun's solar wind is equalled by the very low interstellar gas pressure.
The Voyager 1 robotic spacecraft is currently passing through the heliopause and has measured its distance at 122 au, making the diameter of the entire Solar System twice this much, i.e. 244 au.
Other definitions put the boundary farther out, somewhere around the Oort cloud (where we think most comets come from). The inner edge of the Oort cloud is probably about 1000 au from the Sun, with the outer edge being around 100,000 au. These numbers would make the Solars System's diameter 2000 au (300 billion km) and 200,000 au (30 quadrillion km) respectively.
Distances to the edge of the Solar System (logarithmic scale showing astronomical units (au))