Successive wins over Chelsea and Manchester United had raised hopes that David Moyes's side could make a late challenge for the final Champions League spot.
Indeed, victory at White Hart Lane on Sunday would see Everton close the gap on fourth-placed Tottenham to five points with a game in hand.
Dejected: Louis Saha (right) laments Everton's European exit as Sporting players celebrate in the background
But with Manchester City, Liverpool and Aston Villa also ahead of them in the chase, captain Phil Neville accepts it is asking too much of his team.
'If you had asked me after we beat United on Saturday, before I had seen the table, I would have said that fourth or fifth place was still within our grasp,' said Neville.
'But we still have a mountain to climb to close the gap on Tottenham.
'Even the gap between ourselves and Villa is seven points. That's three games. When you get to this stage of the season the clock is ticking and you can't beat the clock.
'I just hope we don't run out of time. We have to look at our run-in now and to finish in fifth or fourth means we are going to have to win nine of our 12 remaining games - that's the kind of mountain we are going to have to climb.
'Sunday is a massive game. People will have watched our games against Chelsea and Manchester United and seen that we can beat the best.
'You hope that you always get one team that comes up on the rails, but points on the board in my experience are always better than points not on the board.'
Get the ball rolling: Sporting's Miguel Veloso (centre) opens the scoring with a powerful low drive
Neville's admission is all the more understandable considering the manner in which Everton's shortcomings were exposed in the Estadio Jose Alvalade on Thursday night when they were beaten 3-0 by Sporting Lisbon and crashed out of the Europa League 4-2 on aggregate.
They spent Friday in the Portuguese capital reflecting on the defeat before catching a flight to London, but the only positive for Moyes was the return of defender Phil Jagielka as a second-half substitute after 10 months out with a cruciate ligament injury.
Back in action: Phil Jagielka (behind) battles with Liedson in his first appearance for 10 months
Even struggled to see the plus side, however, as Everton let themselves down in Lisbon again just four months after a 5-0 defeat there to Benfica.
He said: 'It's not the way I would have wanted to come back, not with the goals that we conceded.
'I had waited a long time for it. Although we were beaten 3-0, I will look back on this game as the end of a long journey.
'I went through 30-40 minutes pretty much pain free. The result hurt, but in that sense I am happy.
'Europe hasn't seen the best of Everton. We are sick at the sight ofLisbon. We just want to get off home and concentrate on Tottenham.
'Trying to get back into Europe is the only thing we can hold on to.All we can do is hope the big boys win the League Cup and the FA Cupand extend the places for the Europa League next year. We enjoy theoccasions and I'm sure the fans do.'