Arduino Based Solid State Drive I've searched high and low for something to get me on the right track, it's making me think it might not be possible.

Basically, I want to create a cheap SSD using Arduino (or another microcontroller if it is too weak). I know I would have to implement the SATA protocol on the device and an OS driver for the disk drive. Assuming I could jump those two hurdles, what else would bar me from implementing a rudimentary SSD? I'm not even talking about full production quality, just a SATA wire coming from a mother board plugged into a socket on a breadboard. Do I need flash storage or would I need another kind of chip to handle the data needs?
The good news is that SATA drivers are fairly generic so you don't have to write the OS side. The bad news is everything else.

just a SATA wire coming from a mother board plugged into a socket on a breadboard

Let me stop you there. The minimum data rate for SATA is sufficiently high that you need a PCB. If you try and do this with wires or breadboards it will simply not get through; it will be lost to parasitic capacitance.

(Many years ago as a student I tried to do something vaguely similar with a wire-wrap board. At just 20MHz you could see a clean signal go in one side and a noisy unreadable mess come out the other side. Actually seeing it on a scope drove home the importance of shielding.)

Do I need flash storage or would I need another kind of chip to handle the data needs

Where would you put the data otherwise? It might be an entertaining first step to create a read-only drive with constant contents, but it's not very useful.

You need a chip with a SATA interface. This will almost certainly be an ARM and almost certainly not in DIP. I can't remember at the moment whether SATA master and slave "PHY" interfaces are different or whether it's all defined in software, someone else can answer that.