Personally, I'm gutted we're leaving the EU, but fully understand the reasons that many want out. I'd also fully support a 2nd referendum with some clauses stipulating particular vote margins have to be met to activate Article 50. Not only this but how can you decided to leave the EU when almost half the entire country wants to stay, while a percentage of the Leave camp have already admitted regret for their vote, stating it was either based on lies or a protest vote. However I can't see any of the above happening.
All these calls for a 2nd referendum will make for political suicide - any MP or PM pushing for this is basically saying they don't believe in democracy, or don't believe in the version we currently have - a very precarious position for a democratically elected politician. Whoever takes the reins from Cameron, very likely a Leave supporter, is going to avoid this like the plague, and Cameron certainly isn't going to push for it given he's already said, to the effect of, 'why should I do any work towards enacting Article 50?'
Nevertheless, I think the next 2 years are going to be incredibly interesting, if not slightly daunting. I just hope that a strong British attitude to put the best foot forward prevails, and we won't let this decision tear the country apart.