In the number 5.008 g how many significant figures are there?

There are 4 significant figures.

Rules for sig figs:

Rule 1: Non-zero digits are always significant.
Rule 2: Any zeros between two significant digits are significant.
Rule 3: A final zero or trailing zeros in the decimal portion ONLY are significant.

You can find more in depth explanations about these rules here: http://chemistry.bd.psu.edu/jircitano/sigfigs.html