You can weld stainless, you can forge weld stainless, and you can forge stainless...
Forging those blades into a bar is plausible, maybe not practical. Stainless needs to be forge welded in essentially a vacuum, or at least an oxygen free environment. If you were to get a piece of square steel tubing, weld a cap on the end, then stack the blades tightly, or crush them into little pieces, fill the can you created, weld on a lid, airtight. Bring it up to welding heat, let soak at temperature, then use a hydraulic press to compress the can and its contents, you could indeed press yourself a bar of steel. Would it be inclusion free? Maybe... Could you make a straight razor? Sure! Would it be cost effective? Nope!! Environmentally friendly? After forging?? Again probably not. Pride and self satisfaction, plus the bragging rights? Damn straight! (No pun intended! lol!)
OP's question was can it be done... Yes. Yes it can. Is it practical? Way cheaper to buy a nice straight.