Joseph Nasi might have been the modern world's first ever Jewish head of state... and its first ever Zionist. Let's begin.
The state in question is the delightfully-named Duchy of the Archipelago, also known as the Duchy of Naxos. The archipelago lies in the Aegean Sea between Greece in Turkey. From the 14th to the 16th century, it was controlled by the Crispo dynasty of Italian origin.
Nasi was a Portuguese marrano, practicing Judaism in secrecy. After the Inquisition ramped up its persecution, he bounced all over Europe, eventually making his way to the Ottoman Empire, where he engaged in trade and politics. The Ottomans were not exactly fans of Christians ruling Naxos, so in 1566, they took over, made Nasi a duke, and put him in charge.
We're not sure if Nasi was a good ruler or a bad one, but we do know that his Judaism was no longer a secret. Even before his dukedom, he tried to establish a new Jewish state, first in Cyprus and then in Palestine. Jews shouldn't wait for the messiah, Nasi said. Come back to the Holy Land! This was three centuries before modern Zionism.
Alas, Venice and the Ottomans went to war, and that was the end of Jewish settlement. Nasi died in 1579, and Naxos simply got absorbed into the Ottoman Empire.
It was a good try.