This sentence is a bit confusing to me: Za necelé tři roky má začít éra vesmírné turistiky. My guess at a translation would be "In less than three years the era of space tourism will have begun." What's really confusing me here is "má začít"... what's this?
I think the translation could be: in less than three years.... should begin. I'm not 100% sure though.
My proposals: Za necelé tři roky má začít éra vesmírné turistiky. = Má se za to, že za necelé tři roky začne éra vesmírné turistiky. It is supposed, ... (It is expected, ...) ... the era of space tourism is supposed to begin ... ... will begin supposedly ... etc.
Frany was right, it means "should begin". According to my source, mít can sometimes mean "should" (German "sollen").