How far apart are artificial satellites near Earth?

In some locations, they can be a few hundred kilometers apart!

There are about 5000 satellites in orbit out to the geosynchronous orbit distance of 45,000 kilometers. This represents a spherical volume of something like 4/3 pi (45,000km)^3 or 4 x 10^14 cubic kilometers. If we divide this by the number of satellites and then take the cube root, you get a 'scale' of 4,000 kilometers, this represents the average distance between them.

However, for communication satellites and other 'geosynchronous' satellites of which there are probably about 1000, you have to look at a ring with a radius of 45,000 kilometers, and a thickness and height of about 100 kilometers or so. This gives an approximate volume of 100 x 100 x 2 pi (45,000km) = 9 x 10^8 cubic kilometers. Again dividing by the 1000 satellites and taking the cube root of 900,000 you get about 95 kilometers.

Also, there are only a handful of favorite sites in this belt for 'bouncing' signals from Europe to North America, and across the Pacific. Satellites are parked every 0.1 degree along this thin belt which is perhaps only a few hundred kilometers wide at Geosync. This means that there is a severe hazard of satellites colliding with each other unless they are equipped to keep themselves actively 'parked' in their assigned slot. Also, when satellites die or fail, they may be placed into a new orbit which gets them out of the Clarke Belt and therefore keeps that spot open for future working satellites. Although space is vast, there are some parts of it that are very popular and already over crowded!!