The main problem is the Dimona reactor in which Israel's nuclear
warheads are produced. Israel has not signed the 1968 Nuclear
Non-Proliferation Treaty (NPT) and does not allow inspectors in the
facility.
From Nuclear energy and desalination in Israel by Or Rabinowitz:
This facility was never placed under IAEA safeguards or any other
international controls, with the exception of several limited
inspections by American scientists during the 1960s (Cohen
2010). Israel’s refusal to place Dimona under safeguards and open it
for full inspection has been the biggest obstacle to its attempts to
develop nuclear energy.
Since Israel didn't sign the NPT it couldn't import reactors from the US:
Jimmy Carter’s victory brought an end to the Nixon/Ford
initiative. Carter’s administration supported a stringent
nonproliferation policy, embodied in the adoption of the 1978
Nuclear Non-Proliferation Act, which established strict conditions
for nuclear exports. These included the demand that any state
wishing to buy nuclear technology would be required to place all of
its nuclear facilities under IAEA safeguards. Since Israel refused
to place Dimona under safeguards, this meant that it was no longer
eligible to purchase American nuclear reactors.
Another problem is Israel's small size:
Of particular concern is the fact that since Israel is
geographically small, a single nuclear accident could, in theory,
immediately pollute and affect the entire country or large parts of
it, poisoning underground water sources with radioactive pollution,
and contaminating residential and agricultural lands with
radioactive fallout (Grantz 2010). The fear of a major nuclear
malfunction is aggravated by two factors: First, Israel is situated
in the Great Rift Valley, an area prone to earthquakes; and second,
Israel is engaged in an ongoing regional conflict, confronting both
state and non-state actors. Hostile actors might execute an attack
against an Israeli nuclear reactor, despite the fact that Israel’s
Arab neighbors would also be affected.
Not mentioned in the article is that it also has very few locations in
which radioactive waste can be stored. The Negev is likely the best
location for it but is still less than ideal due to the lack of
mountainous terrain. Most other population-wise small countries are
large enough so that both the nuclear plant and the waste can be put
"out of sight, out of mind" but not so with Israel. Then there is the
whole "holy land" thing. A nuclear disaster in Israel would literally
be of biblical proportions.
Israel is dependent on desalination plants for its fresh water supply
and those requires a lot of electricity to run. But recently there has
been several major breakthroughs
in desalination technology making it much more energy efficient so the
need for electricity is decreasing. Plus, Israel's relations with its
neighbours is continually improving so relying on imports of fossil
fuel from the Arab countries is not such a big deal anymore.
Another big problem is that nuclear power doesn't make economic sense
anymore which is why very few new reactors are commissioned. The huge
upfront cost and the limited lifetime of the plant makes solar and
wind energy competitive with it.