Just to address the two most quoted reasons for not holding another referendum:
"the majority voted for leave"
"Another vote is undemocratic"
These are slightly disingenuous and do not address the whole picture. The slight majority of the votes counted were for leave.
51.9% is not an overwhelming majority in any case. For a general election on a term of 5 years - fine. But given that the default is maintaining the status quo, having a majority of a couple of percent for carrying the other 48.1% into this chaos isn't sufficient.
Not only that, the referendum was confusing - lots of the information provided for people to base their vote on was inaccurate, either purposely or inadvertently.
The majority of people polled who didn't vote cited that the information provided was conflicted or confusing and they felt disenfranchised by the experience. A tiny majority in this case does not mean that giving other people the chance to now make an informed decision is undemocratic.
The majority of the country did not vote for leave. 72.1% of the registered electorate voted. The "leave" vote only accounted for 37% of the total electorate. Yes, the other 28% should have made an effort to vote but if many of them felt they had no say, no correct information and were unlikely to make a difference, then the situation may have changed for them and they're more likely to take part in second referendum.
Not only that, other people with a big stake in Britain's place in Europe were unable to vote.
Over 700,000 British citizens who lived in the EU for more than 15 years were denied the vote, despite being able to participate in General Elections.
More than 3,000,000 EU Nationals who've worked in the UK for over 5 years were also denied the right to vote - despite paying tax and supporting our economy throughout that time, many of them with families and their entire livelihood being rooted in the UK.
They have a right to participate in an agreement that will impact their entire lives and split many families up dependent on the immigration agreements that are finally decided on.
Lastly, if you take the rest of the population into account - the people who have their entire lives rooted here but are unable to vote for various reasons - these people are having this immense decision made for them based on just 26.5% of the population's wishes.
Once again - staying in the EU with the current economy configured to work with it in this way - that's the default. Even if you take the tiny majority of the voting public into account, how is this more democratic than a second vote - most of the people in the country didn't vote for leaving the EU.
Even without all this, there's strong arguments to say the original referendum was undemocratic. There should have been a significant margin either way to make so eventful a decision and it should have been carried out after the first referendum left the country so divided.