The use of referenda decide questions of borders and sovereignty is not unreasonable - and I have no strong feelings about Scottish and Catalan independence, to take two topical examples - but the idea that fundamental changes can be made based on a bare majority of (say) a 70% turnout of voters seems insane to me.
A 52% vote (to pick a not entirely random example percentage) under these circumstances might represent little more than a third of the population, which hardly constitutes mass support. Depending on the political system, such support might not even translate into the ability to form and maintain a government. Such a plurality is also vulnerable to even small changes in sentiment (conceivably driven by something as ephemeral as the weather) and could easily lead to unstable flip-flopping. And any small majority is invariably going to produce potential irredentist regions, with majorities, potentially quite large, against separation. What happens, for example, if Barcelona votes strongly against independence in a successful Catalan referendum, or Orkney and the Shetlands vote overwhelmingly against in a Scottish independence referendum?
So what should the threshold be? One possibility would be to require some kind of super-majority, perhaps 60% or 66.66% of those voting. If that seems anti-democratic, an alternative would be to require a majority of eligible rather than actual voters. This is effectively the situation in Australia, which has compulsory voting. Australian constitutional referenda also recognise the asymmetry involved in making big changes: the wording is always as a possible change to the constitution, not as a simple selection between two options. (Australia also has some protection for its component states, with constitutional referenda requiring a majority in 4 of 6 states to pass.) Curiously, there's been some muttering about Western Australian independence recently...