Arduino Based Solid State Drive
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.