Can a 4X8 beam support a floor with a span of 13 feet?

Technically Yes, but you'll need to check with your building inspector because there are codes regarding structure that vary from place to place, and the inspector isn't going to pass something based on my say so.
The old rule of thumb was 1 inch for 1 foot, so a 12 foot span would require a 2 x 12.
A 4 x 8 isn't equal to a 2 X 16, or even 2 2 X 8's, but it should do if the load isn't too great.
I'm guessing you want to go with 8" stock because you've got some headroom issues. You'd be better off structurally with an LVL (Laminated Veneer Lumber) or OSB (Oriented strand Beam) or a truss, but these are kind of pricey.
If you get in trouble see if they'll let you go with them on 12" centers.