Ok- Lets blow up a balloon and lets say our galaxy as we know it and all we know is inside that balloon— then whats beyond the ballon (or whats holding up the balloon)and beyond the next— it must keep going forever— there is no beginning of space and no end to it.
What is beyond the balloon (universe)?......Nothing.....Aristotle, when asked what is nothing,....he replied...."Nothing...is what rocks dream of...No thing. Space is a thing...there is no space beyond the spacial boundary....dimentionally speaking. It is a very hard concept. Think of it this way...the universe began at a moment and at that moment time, space, energy, matter....all came to be. The explosion of creation was a release of energy on a scale which we simply cannot comprehend, and from that there came to be 10 to the 80th atomic particles. Also space came to be. We know that at the pressures and temperature the character of matter was unlike anything we know of now and that the rate of expansion at Plank time was unimaginable. But it expanded at a rate. Then 2 minutes later the expansion continued at an incredible rate, creating space. And that has been occurring since the explosion....and space has been being created. That barrier continues to expand at a rate faster than the speed of light. Matter and energy are fixed (2nd thermal law). So as we speak ther is not a condition where space goes on forever. There is a barrier (constantly expanding).
Will it go on forever....No.....eventually, without intervention, will die a cold death, with all atoms, subatomic particles, being washed out into a cosmic ocean of near nothingness, approaching absolute zero....entropy will have completed its mission. But it will titrate to an end, and space will cease to expland(or nearly so).
If you want to read more on the subject look at Kaalams Cosmological Arguemnt, The Cosmological Arguements (both of these arguements are families of arguements explaining these matters)