Brian Monetti, I <3 astronomy
With a lever large enough! In fact, it is possible, but Earth would likely not fare too well, and the new 'suns' would not resemble the existing one we have. Here is one way it could happen:
An extremely large black hole (or two) swings right past our solar system, and its event horizon passes within just a few million miles of the sun. Earth would be pulled in and fall forever, but parts of the sun might survive. In fact, if the black hole was traveling fast enough (probably a considerable percentage of the speed of light,) it would stretch the sun like silly putty as it flew by. In fact, two orbiting black holes might increase our likelihood of success, but that geometry is a bit too hard to explain in this text field! Anyway, the 'sun' would be reduced to a long bubbling trail of hot gas, no longer supporting nuclear reactions and only glowing because of leftover heat. It would become a nebula, and would probably be quite beautiful if you happened to be several dozen light years away. Despite this solar predicament, the gas itself still has gravity, and if it is dense enough, the gas will start to collapse. Depending on how long this string of gas is (probably on the order of a few hundred million miles) and its density distribution, it might collapse into multiple spheres of hot gas, as opposed to the individual one it came from. This gas, coming from the sun, is mostly composed of hydrogen, which happens to be great star forming material. If enough of this gas gets pulled together, a nuclear ignition will occur, and suddenly, a new sun! And if we are lucky enough, there might even be multiple suns. This would also require the spheres of gas to be moving away from each other rather than recombine into one, but with a little luck that might just happen.
Unfortunately, the new 'suns' would not resemble the old one. They would probably look something more like Wolf359, a red dwarf star that only has 1/10 of the suns mass. It also happens to be very dim, producing only about .1% of the suns light. But, because it supports nuclear reaction, it is considered a star, although a somewhat unimpressive one.
So it looks like it will work! If you want to watch the process happen, you better build a very fast spaceship and pack a few lunches, because the process would likely take a few thousand years.