In the world of Dragon Ball Z, can you stay in Super Saiyan form all the time? The answer is yes, but with a few catches. Making Super Saiyan one's base form was part of Goku and Gohan's training leading up to the Cell Games, and Vegeta proved he could achieve such power as well. However, as the Super Saiyan transformation evolved and new forms emerged over the course of the series, the answer to that question became a lot more complicated.
Goku and Gohan's Super Saiyan Training During the Cell Games
Prior to the Cell Games, Goku, Gohan, Vegeta and Trunks trained in the Hyperbolic Time Chamber in groups of two. While Vegeta and Trunks tried to find a form beyond the Super Saiyan in order to make their power comparable to Trunks, Goku and Gohan realized that this really wouldn't help them all too much, since pushing their power further would give them increased strength but reduced speed and dexterity. Instead, they pushed to make the Super Saiyan form into their base form.
Super Saiyan transformations require extensive amounts of power to maintain. Making Super Saiyan one's base form saves all the energy wasted on transformations. Saving this energy allows one to put even more power into their Super Saiyan form, increasing battle potential exponentially. Goku and Gohan spent an uncertain amount of time as Super Saiyans without ever reverting back, possibly several months in the Hyperbolic Time Chamber. This is referred to as a Full Power Super Saiyan.
Becoming a Full Power Super Saiyan became a requisite for further transformations. Gohan couldn't become a Super Saiyan 2 until he made Super Saiyan his base form. It's implied that any advanced Super Saiyan transformation requires first achieving the Full Power Super Saiyan form, meaning Vegeta, who later achieved Super Saiyan 2, is also capable of making Super Saiyan his base form.
Can You Spend a Lifetime as a Super Saiyan 2 or 3?
So Goku, Gohan and Vegeta could remain in their Super Saiyan forms for extended periods -- and do, as part of their training. However, could they spend a lifetime as a Super Saiyan 2 or 3? While one might think the same principle would apply, it doesn't exactly.
The original three Super Saiyan transformations are not separate transformations, but rather hyped-up versions of the same core transformations. This is why Super Saiyan 3 burns so much of Goku's energy: because he's pushing more energy through his system than he normally should be able to. This is also why Goku's Super Saiyan form becomes stronger as he trains. The stronger he is, the stronger his other forms become.
By its very nature, Super Saiyan 2 and 3 can only occur if the Saiyan in question is exerting extra effort. Much like the Kaio-Ken Technique, Super Saiyan 2 and 3 cost a large sum of energy to maintain. By maintaining that form, Goku would only be increasing his base effort, allowing all his forms to become stronger. However, by the nature of the technique, Goku can't turn Super Saiyan 2 or 3 into a base form. Of course, the strength that a permanent Super Saiyan form provides makes these additional levels unnecessary.
You CAN Spend a Lifetime as a Super Saiyan God or Super Saiyan 4
Almost paradoxically, it's actually easier to make a base form out of the far more powerful Super Saiyan God and Super Saiyan Blue forms. Super Saiyan God permanently transforms the person who channels that power. Upon transforming into a Super Saiyan God, their power is forever imprinted on the transformed Saiyan. All it takes is one transformation into a Super Saiyan God to make the form a default.
Super Saiyan Blue is what occurs when you transform into a Super Saiyan while a Super Saiyan God. By that logic, then yes, Goku and Vegeta could undergo the same training Goku and Gohan went through during the Cell Saga and try to turn Super Saiyan Blue into their base form.
Similarly, Super Saiyan 4 functions completely differently than Super Saiyan 3 and doesn't burn energy in the same way. In Dragon Ball GT, Goku can barely maintain his Super Saiyan 3 form, but he comfortably maintains his Super Saiyan 4 form like a normal Super Saiyan form. It makes sense that it would function closer to the normal form in allowing one to stay transformed for extended periods of time.