Liverpool - Solskjaer sorry after 'unworthy' United slump to Everton defeat
Among many areas of criticism, the desire and fitness of United's players was questioned by their manager, who was equally baffled when asked whether his team had been shamed by the performance.
"I don't know, you've got to ask them," he said. "I've said to them, you're not going to get the answer from me but if you want to play at this club, it has to mean more.
"I want them to be the hardest working team, that's what we were under Sir Alex (Ferguson). You ran more than everyone, every single week, a shift up and down that pitch for your team-mates.
"We can't change the whole squad, it has to be one step at a time but I said all along, I'm going to be successful here and there are players who won't be part of that successful team."
Everton took the lead after 12 minutes when Lucas Digne launched a long throw deep into the United area, Dominic Calvert-Lewin helped the ball on with the flick of his head and Richarlison was well placed to send a spectacular scissor kick finish flying past David de Gea.
After a humbling 3-0 Champions League exit at Barcelona in midweek, it was not the start Solskjaer had been looking for and it did not take long for Everton to make the situation worse.
After 28 minutes, Everton cleared a United corner and immediately broke on the counter-attack, with winger Bernard playing a delightful ball inside to Idrissa Gueye.
He, in turn, released Sigurdsson who, with Nemanja Matic curiously backing off and allowing him space, unleashed a magnificent 25-yard shot which bounced just in front of the diving de Gea on its way into the bottom corner.
And the game was effectively killed off in the 56th minute when a Sigurdsson corner was punched clear by de Gea and the unmarked Digne sent a magnificent, first-time volley flying back past the United keeper from 20 yards.
Even though Richarlison had limped off through injury, the agony was not over for United with the Brazilian's replacement Walcott completing the rout in the 64th minute.
Sigurdsson again carved out the opening with a brilliantly-timed through ball which allowed the former England international the space to sprint into the area and beat the advancing de Gea.
Not until the 86th minute, and a tame Anthony Martial shot directly at the keeper, did United even produce a shot on target.
"That performance is not good enough for a Manchester United team, all the way from me to the players," added Solskjaer.
"We know we let the fans and the club down today because that performance is difficult to describe because it's so bad, they beat us on all the basics."