Is it legal? We have no idea.
The temptation to ride vehicles which are not street legal around on the street in the middle of the city is a strong one, we understand this. But the temptation must be especially strong in Greece, as that is where the following video comes from. It shows a go-kart, plus several other vehicles, being driven around the city very quickly (for a go-kart) doing a bit of sliding around as well.
Be warned, it seems that a GoPro doesn't take terribly stable video on a kart as compared to other surfaces, so it is more than a little shaky.