I'm trying to find a way by boat from Athens to Tel Aviv and back.
I understand air travel would be cheaper and easier - but I wish to do it more for the experience.
There are probably no (ordinary) ferries going to Israel.
Some summers has a ferry from Greece via Cyprus to Haifa. However, from the Cyprus ferries page at Varianos travel:
However, the situation changes year by year, so it's possible that there will be a new operator trying to open ferry lines.
P.S. You could try to board a Ship to Gaza — the Israeli military may be happy to show you some of the less touristic parts of Israel before putting you on a flight home!
P.P.S. Presently, the overland option via Syria may not be a good idea either.