You can record to another drive across a network a.k.a recording to a server for data storage (if setup properly), I wouldn't suggest this (as there are less expensive ways to conquer this). I would recommend purchasing another hard drive(or SSD), and recording to that.
PCPartPicker part list /
Price breakdown by merchant
Storage: Samsung 850 EVO-Series 250GB 2.5" Solid State Drive ($97.99 @ Amazon)
Total: $97.99
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-06-08 14:32 EDT-0400
Or
PCPartPicker part list /
Price breakdown by merchant
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($48.75 @ OutletPC)
Total: $48.75
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-06-08 14:32 EDT-0400
I recommend the HDD, since you're playing on another, you can have the hard drives full potential, and it shouldn't slow your PC down no where near as much. But if you want to risk slowing down over storage space, choose the SSD. (They're so much faster)
Now for streaming, that's completely different. You're going to have to play with the settings yourself. I do recommend using OBS (Open Broadcasting Software) for that, it's less resource intensive. Make sure you have decent internet!
https://obsproject.com/estimator <-- Use this to see what settings you should be at.
P.S. Don't bump your topic 2 hours later.
A lot of people don't take kindly to it.