There are 4 things that control the exposure of your photos. Shutter speed, aperture & ISO...those are the on-camera controls we have. The 4th one is the amount of light (whether ambient or added by you (flash)).
So if you can't add light to the indoor space, and you can't or don't want to use flash...then you have to use your exposure settings on the camera. When you want to freeze the movement of fast subjects, you need a fast shutter speed, so that won't help you in terms of exposure. The only other options, as mentioned, are to increase the ISO or shoot with a larger aperture.
I don't see that you've mentioned what camera/lens you are shooting with, but I'd guess that you're already at your maximum aperture. So the only option for a larger one, would be to purchase a lens with a larger max aperture...if you have a camera with interchangeable lenses.
The other option is to use a higher ISO. But of course, that will mean more digital noise. That's the trade off that you make.
And lastly, sometimes you just have to realize when you are at the limits of your gear.
In an ideal world, you could put some high powered and fast cycling strobe (flash) lights into the building and trigger them when the camera fires. This is often what pro sports photographers do for indoor sports like basketball & hockey.