"Messi and Cristiano [Ronaldo] have been the best players in the world for the past decade", said Solskjaer.
Ajax stunned Ronaldo's Juventus 2-1 in Turin to progress to the semi-finals in the day's other match.
A run in the Europa League might cost the club financially, but that is a more realistic level for where United are at the moment.
Then, Barcelona had their own eight-minute flurry as United nearly conceded a penalty, ruled out after Fred took the ball before bringing down Ivan Rakitic, and then did concede, twice, to Messi.
90 - Ninety per cent of Manchester United's Champions League goals this season have come away from home. Donny van de Beek cancelled out Ronaldo's opener before Matthijs de Ligt took the victor to let the Dutch visitors advance 3-2 overall.
Four minutes later, the tie was over, as Messi's tame right-footed effort somehow found its way under David de Gea to make it 3-0 on aggregate and hand Manchester United an impossible task.
Messi got his double soon with more than a little help from De Gea.
United began positively with Rashford prodding a shot against the bar inside one minute, but they cultivated their own downfall within the space of four minutes.
"It could have been different if we had started better and we got the first goal, but now we have to concentrate on next week".
"The quality of their finishing was absolutely outstanding". He then turned Phil Jones inside out on the stroke of half-time to release Jordi Alba.
The contest was firmly put to bed when Coutinho was allowed to turn to his favourite angle and curl his trademark shot past De Gea, making the score 3-0.
United are yet to win the Champions League since 2008, back when Sir Alex Ferguson was in charge. The Portuguese forward has also scored 51 per cent of his total goals in the competition in the knockout stages (64/125).