If I have a 100mm lens, how far away can I shoot the subject? If I am shooting basketball, how far can I shoot the subject and still get a sharp picture?
If you are on the baseline or sideline, near the action, you probably want a 35mm or 50mm, and something between 80-150mm for longer range shots, depending on whether you want a full or half body shot.
Not sure what you mean about "how far can I shoot the subject and still get a sharp picture". Sharpness depends on how well you focus obviously, and how little camera or subject motion blur you get. So you want a fast shutter speed, at least 1/125 or faster, to prevent blur. With indoor lighting, that's hard to get without flash or a very fast expensive lens
As long as your subject is further than the minimum focus distance of your lens, you will be able to get a sharp picture.
There are other things that affect sharpness, including subject motion and depth-of-field. Since depth-of-field increases with distance, the further you are the more chances of your subject being in focus as it moves while you are focusing and taking a photo.
Subject motion is not controllable by you but the wider your aperture, the faster the shutter-speed to get a consistent exposure, this increases your change of freezing the motion of your subject. The only catch is that lenses are always softer wide-open, if you have a cheap 100mm lens this may be severe but if you have a high-end one then it should be minimal.