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If I remember correctly in one Episode of Stargate SG-1, the president of the United States declares some state of national emergency (world was attacked by aliens) and was able to suspend elections, weaseling around the 22nd Amendment and being able to stay in office indefinitely by keeping up the emergency state.

Is it actually possible for the president stay in office longer than the usual four years or even forever, by using some (basically) legal tricks?

The short answer is no. The longer answer is that this framing isn't particularly helpful.

There are a number of overlapping factors that prevent the president from legally suspending elections like some tinpot dictator. In particular Article II Section I of the Constitution and the 12th, 20th, 22nd, and 25th amendments which combine to define presidential elections and succession.

Terms ending are not directly linked to elections being held. Even if elections could not be held for some catastrophic reason, the president should still leave office at the end of their term according to the line of succession (if no elections occurred at all many of their terms will have ended at the same time, but there should be a President Pro Tempore of the Senate because Senate terms are staggered). This is all theoretical since it's never been tested and hopefully never will be.

Perhaps an even more boring reason why the president cannot simply suspend presidential elections is that the federal government does not run elections. The state governments do. A state government may have the ability to suspend or reschedule an election, and there are rules in place for if a state fails to make a selection on the prescribed day. If they still haven't named electors in time for the meeting of the Electoral College, that state will simply not cast any votes.

The only legal way around any of this would be modifying the Constitution. Because of course you can make anything 'legal' if you change the definition of what 'legal' means.

At the end of the day though, the real world isn't a game. Even if you could find some obscure and bizarre legal loophole to override precedent, the law isn't a set of magic rules. Its just rules that we made up. Nobody would have to accept it. A democratic government governs with the consent of the people. Attempting to use force to subvert the will of the people would be how a president becomes a dictator.

The Constitution sets a presidential term at 4 years, and the 22nd amendment pretty firmly sets a two-term maximum:

No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once...

Since this was a Constitutional amendment, there is no legal way to extend a presidential term after 8 years (or technically 10 if you were a VP-turned-Pres mid-term) without passing a new amendment to allow it. This applies even if elections themselves are suspended for an emergency - the president's term is up when it's up, regardless of whether there's anyone else to take up the office.

Whether or not there's a illegal but effective way to stay president after two terms would be pure speculation.