Safety regulations for take off spectators around airports?
Most airports are intentionally built away from other areas because of practical problems like noise and potential structures getting in the way. There's also the rather significant requirements to build a runway itself (long, relatively flat and level). But there doesn't seem to be any hard minimum safety distances. It seems to be left up to the airport itself to determine safe distances. 

One of the best examples I can think of (with limited clearance around the airport) is Princess Juliana International Airport in St. Maarten. The runway approach passes directly over Maho Beach, a road, and then you have a 7000ft runway (the largest flat spot the island can muster). In the picture below, the fence marks the end of the airport.



It produces some spectacular videos of planes passing less than 100ft over the beach and people watching on the beach are impacted by the blast of jets, who must stop at the end and bring their engines up to speed before starting their roll.

The incident mentioned happened at Skiathos Airport, which is also an island airport with limited land for a runway (shorter than St. Maarten at 5300ft).

So it seems to be that if you have a long, flat, and relatively level spot to build a runway, there's nothing to prevent you from building an airport there.