Defeat against Sunderland left David Moyes' team trailing fourth-placed Chelsea by six points having played a game more than the London club.
Distin, however, feels that it is unrealistic to expect Everton to regularly beat the likes of Tottenham and Arsenal given their respective budgets.
"I don't think people talking about the Champions League or not changes much for us on the pitch," the centre-back told reporters.
|1/3||Everton are 1/3 with BetVictor to finish in the top six
"We all dream but we have to keep in mind as well that we fight with such a small squad.
"When we spend £1 million, the teams we compete against spend £20m. You have to remember that. It has been a really good season.
"Clearly at the end of the season when you play three games a week it's a bit difficult, especially for a team like us where we have the same starting 11 most weeks.
"We don't have a big squad. But we never hide behind this excuse. Yes, it's difficult but it's not the excuse for why we lost."
Distin has now turned his attention towards securing a place in the Europa League and is determined to end the season on a high regardless of the club's finishing position.
"Winning all our games would have meant we could have kept dreaming [about the Champions League] and obviously now the dream is going a bit far away," he admitted.
"We'll see about the Europa League now but no matter what happens, we want to win every remaining game until the end of the season. We missed a good chance against Sunderland but there's always the next game."