Now this is thinking big... but not clearly.

[unixronin]

(Read the comic here)

There's just one problem with this idea:  If you take a star apart, it stops working.  Then all you have is a whole lot of hydrogen, helium, and relatively limited amounts of heavier fusion products.  The hydrogen you can get far more cheaply and easily, and in more than adequate quantities, from a gas-giant planet; the heavier elements you can mine far more cheaply and easily from asteroids.  It's a pretty poor return for an engineering product of such magnitude -- and for what, really?

If you have the technological capability to take apart a star, you probably have the capability to build a Dyson sphere (or, for that matter, a buuthandi, which is conceptually similar but built on a slightly different principle known as the Dyson bubble), which enables you to actually capture and use all of that energy.  OK, so maybe the star only remains usable (and the sphere only remains habitable) for another five billion years or so ...  You know what?  Honestly, the time to start worrying about that is probably at least about 4.9 billion years from now (which still leaves you about a hundred million years to do something about it).  If, that is, you're still around in 4.9 billion years and still dependent upon individual stars at all.  Odds are, one or the other of those will turn out not to be the case.