Maybe things have changed since Gateway bought Emachines, but it used to be that they were infamous for creating systems with absolutely no upgrade abilities. So provided you have some means of mounting the old drive in the new system, there shouldn't be a problem.

Of course my guess would be that you could fix the old system by simply giving it a new power supply. It's a VERY common part to fail in computers, and it's also relatively cheap to replace. The annoying part is connecting all the internal plugs. It's especially annoying on cheap and cramped cases like you often find with brand name systems.