Actually that headline could apply to Arsenal in two separate ways: the end of the line financially which occurred on 13 June 1910, and the end of the line for Billy Wright.
The 1910 tale has been followed in great depth on this site through Henry Norris at the Arsenal: There is a full index to the series here. But in essence on this day in 1910 the club declared that it could not continue for another season in its current situation, and proposed that the club should continue by merging with second division Fulham, but with the resultant club playing in the 1st division. The League say that the clubs could merge, and could play where they wished, but it was up to the League to decide where the club would play, and their decision was, the second division.
Of course matters were resolved, first by Henry Norris paying off all the debts, and second by then paying for the building of a new stadium in north London which instantly made Arsenal one of the best supported clubs in the country.
As for Wright, he was a total unmitigated disaster of a manager with league finishes of 7th, 8th, 13th and 14th, and the lowest ever crowd for a league game at Highbury all meant he has been rightly consigned to the dustbin of football managerial history as an unmitigated disaster. He finally did the right thing and resigned on this day. The full story is told in the wonderful book “Arsenal the long sleep” by John Sowman, copies of which are available from the author.
13 June 1910: Woolwich Arsenal came to within five minutes of extinction at the AGM of the Football League, as the League executive turned down the idea of Arsenal merging with Fulham and playing in the first division. See also here; and again here. For a detailed account of the month’s proceedings see here.
13 June 1917: Henry Norris attended Buckingham Palace to be knighted by King George V in the midst of the first ever daytime bombing raid over London!
13 June 1966: Billy Wright finally resigned. He was the first Arsenal manager since Chapman who was appointed to the club without having had a previous link with Arsenal and was by and large an unmitigated disaster.
13 June 1980: Having signed the day before, Clive Allen at the age of 19 was now heralded as the most expensive teenager in Europe in joining Arsenal from Queens Park Rangers for £1,250,000.
13 June 1983: John Devine joined Norwich on a free transfer having been out of the side with a long-term injury. He had played 112 league games for Arsenal and continued playing for another eight years. He later worked for the Football Association of Ireland as a Tutor/Assessor and is currently Technical Director for Almaden Soccer in the Silicon Valley.
13 June 1985: Fifa’s ban on all English teams playing overseas was introduced following the Heysel riot by Liverpool supporters This was subsequently modified to allow friendly matches by English teams.
13 June 1988: Steve Bould signed for Arsenal for £390,000. He had played 183 games for Stoke City and nine on loan with Torquay. Both Arsenal and Everton made offers. He made the right choice.
13 June 2008: Aaron Ramsey signed from Cardiff. Both Man U and Arsenal bid for him, but he chose Arsenal undoubtedly in part because of the connection between Arsenal and Cardiff via Terry Burton. He finally left in the summer of 2019 as his contract came to an end.
13 June 2014: After rejecting the offer of a new contract Bacary Sagna moved to Manchester City on a free transfer. He had played 213 league games for Arsenal in a seven year spell, but made only a handful of appearances for Man C. He was last heard of playing for Montreal in Major League Soccer.
13 June 2014: It was said in some sources that Arsenal sold Chelsea their rights to the re-signing of Cesc Fabregas to Chelsea for approx £5.6m. Arsene Wenger later reported that he had no contact with Fabregas at this time.