United's success in the pre-Abramovich era (say roughly 1992-2003) was the perfect combination of a number of circumstances.
Firstly, Old Trafford was far bigger than any other top-flight ground, especially after capacities were reduced post-Taylor Report, meaning that their matchday income was significantly greater than their competitors. They were always one of the best-supported clubs in the country, but this reinforced it in their favour while simultaneously harming other big clubs with smaller grounds (like Liverpool, Everton, Spurs and Arsenal - the other leading clubs in the breakaway from the Football League which formed the Premier League). They could therefore afford to regularly cherry-pick the best players from other Premier League teams (Eric Cantona from Leeds, Roy Keane from Nottingham Forest and Andy Cole from Newcastle being two early examples).
Secondly, their commercial and merchandising operation was ahead of its time - they were the first to spot the opportunities in having an Official Auto Partner in Japan or whatever it was. They realised that there was money to be made by changing your home and away kits every season. And they were prepared to travel long-haul pre-season in order to grow their fanbase in the US and Asia.
Thirdly, they had an outstanding crop of young players, all of whom were the backbone of their sides for a decade (Scholes, Giggs, the Neville brothers, Beckham, Butt and other squad players), meaning that they had a very settled side with a core of players who'd been brought up together in an atmosphere of success.
And finally, they were managed by one of the greatest managers ever to work in English football, who had the good fortune to be allowed almost free rein by his chairmen throughout his reign.
It's hard to imagine that all of those things will ever coincide in one club again. I'm sure Man Utd will win the League again, but it's hard to imagine they'(or anyone else) will dominate for nearly two decades as they did under Alex Ferguson.