Hamburg's 1-1 draw with Wolfsburg left them three points adrift of sixth-placed Freiburg with two games of the season remaining.
A win would have given the northern German club hope of catching either Freiburg or Eintracht Frankfurt in the race for European places, but Fink realises his side simply are not yet ready for a return to international football.
"I think this was a fair draw," he said. "Both teams played an open game and wanted to win and I'm pleased with the way we fought, but technically it wasn't very good, there's no doubt about it.
"Of course it's going to be even tougher now to reach the Europa League.
"We are heading in the right direction in terms of stability, but we are not mature enough for the Europa League.
"We want to take this club right to the top, but you can only do that with continuity."
Continuity has been Hamburg's bugbear for the past two years with dramatic falls being followed by equally dramatic rises.
Yet whenever they are within touching distance of achieving something substantial, they appear to stutter.
"We need to be more consequential," said Fink to Sky television. "We need players who are willing to go the extra mile for this club."