All games are "almost unlimited" in size. At the start, the only systems that are actually generated are the home systems of the player(s) and NPRs. All other systems are only generated when somebody goes through an 'unexplored' jump point. What a galaxy size of 1000 (for example) does, is make the system on the other side of that jump point a random number from 1 to 1000. If that random number corresponds to an existing system then it links to an unexplored jump point if one is available, other wise a new point is created on that side.
The 'Local Sys Chance' is that that random number is within the 'Local Sys Spread' from the number of the originating system - if set to 75% and spread 20, then exploring from system #555 has a seventy-five percent chance of reaching a randomly determined system from 535 to 575 (excepting 555 itself - that bug/feature was corrected ages ago). The other twenty-five percent of the time the new system number is randomly selected from all 1000 (or 2000, or whatever) numbers.
That last paragraph gets completely ignored in a 'real stars' game, and instead the new system is picked randomly from the nearest 50 (I think) stars according to astrography. Also, since there are no nebulae or black holes or other fun 'space terrain' near us, there is no chance of encountering such in a real stars game.
And any game can start in the Sol system if you wish - it's one of the options given on the 'Star a New Game' screen at the beginning.