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So this morning I turn on the Olympics - and discover that my proposed sport exists already - and it's called Handball. It actually looks like a cross between Basketball and Soccer. (At first, I thought I was watching a basketball game with a really small ball, and a relaxed attitude towards "traveling".)
Judging from the Ali-G inspired names of its players, it's big in the Eastern Bloc. And according to Wikipedia, it's been around since the 1920's. (Who knew?)
This is the only Handball clip I found on YouTube that I didn't annoy me. It's boring - but you'll get the idea.