This would likely require completely new forms of physics and interacting with matter I'm afraid. It simultaneously requires extremely high resolution and high penetration (we need to see deep into the brain). You'll need to be able to measure the state of neurons and connections - probably down to the quantum level - and do this real time, and then record the data that you collected.

I suppose you could flood the brain with nano-machines which would latch onto all the bits and pieces and collect the data? But where would they store it? How would we get them all back out again?

I don't think it's possible to do this with our current understanding of physics. This is not a question of needing better technology, but needing a whole new universe with different physics altogether.

I'm not even sure what is more far fetched, this or superluminal travel. I'm actually leaning towards the former :D