I would usually say, "five point six repeating." But as I am an airline pilot and over the radio we say "decimal" rather than "point" sometimes I will say "five decimal six repeating." Most people probably say "point" or even "five dot six repeating." I think "decimal" is a technical thing that probably only a few people say.
I've also heard people say "five point six six six six six six" (where they say many "six's" quickly to give the impression that they continue forever). And I've also heard people say "five point six six six dot dot dot" to describe the way that it is literally written: 5.666...
I think that "five point six recurring" is also common, but I feel like "recurring" is more popular in Britain, and "repeating" is more popular in the US. As a native english speaker, I would easily understand any of these. Or even "five and two thirds."
Incidentally, I don't think I've ever seen it written the way you typed it, using parentheses to indicate the repeated decimal part. I've only seen a bar above the numbers or the ellipses symbol (the three periods: "..."). Is that a common way to write the repeated decimal in Russia?