I think the proper answer would involve some political points in the past 20 years, so we'll have to ignore those because they're outside of the scope.
Instead, I think we should focus on a few key points: post-1948, post-1973, and the 1990s. I think those are the two moments closest to peace that don't reach into the 20 year rule, though I will discuss a common concept, namely that around the mid-1990s peace was close but disrupted by the assassination of Israeli Prime Minister Yitzhak Rabin.
Post-1948 is the initial, important stage. It was early on in the conflict, and it was a moment where entrenched difficulties in the conflict still seemed somewhat resolvable. Conflicts around the globe were sparking and being solved, particularly related to decolonization during the breakup of the British Empire (see, e.g., India/Pakistan). WWII had left superpowers involved and engaged in international affairs, and the UN was both new and untested, and the Korean War and its difficulties for the UN were yet to occur, while the superpowers also were in a more cooperative stance on a general stage.
It was in this context that the Israeli victory in the war of 1948 with its Arab neighbors was turned into ceasefire negotiations. However, the ceasefire negotiations could easily have turned into a peace agreement. How effective it would've been is unclear, but there were certainly discussions to that effect.
Following the implementation of a ceasefire on January 7, 1949, Israel and Egypt launched armistice discussions on the Greek island of Rhodes, mediated by Ralph Bunche, a UN acting mediator for Palestine (referred to here as the former British Mandate, not as a state, which is commonly misunderstood). Israel felt a strong need to negotiate first with Egypt, and also to make some form of armistice or peace deal, because Egypt was a large and leading Arab power, and failure would likely mean other states would not join in any other armistice agreements. That would leave either continued war or a UN Security Council action (if one could be agreed to), and both presented serious risks to Israel that it was unwilling to take. It was believed that a deal with Egypt meant cementing Israel's existence. Peace, of course, would do that even better than an armistice. After a 6 week negotiation, an armistice was signed, and Lebanon, Jordan, and finally Syria followed, in that order.
By July 1949, the Syrian armistice agreement was signed. Many argued that throughout these negotiations, peace was almost impossible to reach, and that's likely the case. However, you asked about the closest moments, and this was certainly one of them. During the armistice negotiations, there were discussions and contacts that suggested Egypt might accept peace if Israel gave up the Negev, but nothing formal (and this was out of the question for Israel, as that was more than 30% of its territory). Syria's President Hosni Za'im presented a potential peace in exchange for much of the Jordan River and half of the Sea of Galilee, but this too was likely well outside of Israel's abilities to accept, and Za'im was not long in power anyways, and he was executed within a few months of taking power.
Jordan, as before the war, was the most interested in avoiding further conflict. However, Jordan too was interested in peace deals that Israel was unwilling to accept involving substantial giving up of territory. Israel, for its part, believed that having won a war where it was invaded after declaring independence, should at the very least have the right to have peace within the armistice lines without question, because it did not take any territory that any of the other states even had a legitimate claim to. In essence, they argued that Arab states demanding Israeli territory for peace were trying to take territory they didn't own, never did own, and which Israel won by defense, and there was no reason to hand it to Arab states for the Arab states to accept a peace required by the UN Charter.
Nevertheless, the failure of peace in each of these bilateral talks didn't mean the end of peace attempts. The most famous was a multilateral peace conference in Lausanne, Switzerland, convened in 1949 by the Palestine Conciliation Commission, which was set up by the UN General Assembly in December 1948 and run by an American chairman. The peace conference failed miserably. Arab states demanded Israel take in the roughly 700,000 Palestinians who fled or were expelled during the course of the war, while Israel demanded that the Arab states be responsible for integrating many of them into their own states, offering to take back a total of 100,000. Had Israel taken back in 700,000, they argued, the state would've been incredibly unstable; 700,000 individuals who largely opposed the existence of a state, moving alongside (at the time) roughly 800,000 Jews and 150,000 Palestinians already, would mean an Arab-majority state. The Jewish state would be either undemocratic, or die before it was born demographically. Israel also pointed out that it was taking in floods of Jews fleeing or being expelled from Arab states. The Arab states also demanded that Israel give up territory, or at least agree that the armistice lines should be the basis for any peace deal, meaning Israel would need to accept that it would make territorial concessions for peace (essentially, trying to force Israel to accept the concept of giving up land, without specifics yet). Israel refused, and the US tried to break the deadlock on the refugee issue at least by suggesting Israel take 250,000 refugees in. Israel refused, no one made any further progress, Arab states did not move from their demands and Israel did not move from theirs, and peace failed. Still, this was the closest moment post-1948, until over 20 years later.
Continued in a reply to my own comment.