Photography geeks needed!

One simple question: Is it possible at all to shoot good photographs in a place with little light (like a cathedral) without a tripod?

a. with a DSLR
b. with another camera

As I often take pictures of architecture during my travels, I’m getting increasingly desperate because I almost never get good shots from the inside of a cathedral. Most of the times tripods are forbidden there anyway.
I just have a normal standard camera at the moment, and I am thinking of buying a DSLR, but I am not sure if this will solve all the problems I have while shooting.