When Does a Country's External Debt Become a Problem? Here in the UK, the external debt is currently over 300% of GDP. To me this looks very worrying, using my basic financial logic, it seems like this debt will only go up (because of interest combined with the fact we don't have enough money to pay it off). The fact that it's over 3 times the value of the British economy worries me even more, like there's no end in sight...

I understand a country can remain in debt forever (so long as they keep paying their lenders) but how long can a country like the UK keep going like this? Public debt is 94% of GDP so that's just about manageable, but what happens if the external debt keeps rising? or the GDP begins to fall? Can the UK maintain a level of debt that is 300% of GDP forever? Basically, what would it take to turn this debt into a total financial collapse? I've heard of something similar happening in Argentina in the late 90s and they had a catastrophic recession, could that happen in the UK anytime soon?
The position is both worse and better than you describe

If you look at Section 7 and Figure 6 of  https://www.ons.gov.uk/economy/nationalaccounts/balanceofpayments/bulletins/balanceofpayments/quarter1jantomar2018 you will find that UK liabilities (public and private) abroad were estimated to be £10,894.5 billion at the end of March 2018, about 5.30 times UK GDP in the 12 months to March

But at the same time UK assets abroad were estimated to be £10,632.2 billion, about 5.18 times UK GDP 

So the net position here was a liability of £262.3 billion, about 0.13 times UK GDP (a minor rounding issue here), i.e. 13% of annual GDP.  In fact the net position is the difference between two enormous numbers largely resulting from financial activity in the City of London, both of which are estimated with errors, so large revisions to the estimated net position is common  

A small UK net international liability represents the position for many years and so in that sense empirically seems to have been sustainable.  For Greece this net figure became large and unsustainable, contributing to its crisis, and not helped by the Eurozone restricting the possible solutions.  The answer to your question "When Does a Country's External Debt Become a Problem?" seems to be "when people start to worry that they will not be able to get their money back and so stop lending more money".  Turkey may be facing this issue at the moment.  But I doubt there is a fixed figure you can identify