How to mount short handrail with no studs?
A photo would help us narrow down possible solutions, but the bottom line is that you must have at least two solid anchor points. I find it hard to believe that you don't have studs at least on 24" centers. It could be that you're hitting the floor system, but that probably just means that you have solid backing in a horizontal orientation.

If you can't find additional backing, I'd run a flat hardwood 1x4 behind the handrail for the brackets to mount to. It can extend all the way to door trim or wherever is necessary to find backing. Now you have solid backing for your rail brackets.