World of Spectrum has a lot of info on their main site about that sort of stuff, but the basics were a z80 processor and 48k ram with 128k for the +3 model. Not sure of what limitations the NES had, but would settle for Wolfenstein 3D instead.
The first model had 16 or 48k RAM, then the next one had 128k, before the +2 and +3. I had a 48k back in the day.
The thing is, if you want to do palette based graphics with super smooth scrolling and lots of hardware sprites, the Famicom/NES was amazing for 1983, doing all of that with a tiny amount of RAM. The Speccy had no hardware sprites and had to draw the screen manually, so scrolling wasn't smooth and colour bleeding was a fact of life (what little colour it had).
But what the NES can't do well, like KingMike said, is draw an image that it doesn't already have in ROM. With enough ROM space and one of the later mappers you could have fantastic 2D animation with no CPU overhead, but Doom requires a viewport with a constantly-updated screen, which would be a hell of a job for the NES. The Speccy has to draw the screen manually anyway, so it's actually not such a big deal.
Basically the NES was built for shmups and platformers, and does them astonishingly well for the time. But 3D type games are something else. They had Elite but that's a lot simpler, just some simple shapes and pixels. Even the SNES struggled with Doom, don't forget, and needed a Super FX chip to get something close to the PC.