There are many ways this could be done, including multiple pure framing/joinery options some of which wouldn't even require any metal fasteners. These kinds of all-wood joints are best done by pros though since even though this would now fall under "rough carpentry" the fit would have to be at a level far in excess of what's normal for that, approaching that of cabinet work with tolerances well under 1/64" ideally.
A more modern joinery option that first occurred to me is to cut off the ends of the five beams at the correct angle (72°) and then form half-laps on each of them so that they can interlock. The ends would look somewhat like this:
Then drill vertically through the joint and into the top of the post for a bolt which will tie the structure together.
I'd be tempted to just used metal reinforcement plates myself though, with plenty of screws, or bolts for maximum security. Much faster and possibly more secure at the end of the day. If you use a basic T-shaped connector on the inside angles for example these can be easily bent to suit the interior angle. For the outside a basic steel band can be used, again custom bent to fit.
Regardless of the main joining option used some diagonal bracing I think is a must-do, although joining a pentagonal structure to square-section beams at each corner will require some tricky fitting work.