I just changed all the palettes in Castlevania 1 using Stake, but after firing the game up in Nestopia, it looked completely different!
Apparently, each emulator more or less has his own palette...

I found someone who seemed to have put a lot of work in getting an accurate palette and imported that one into Stake but now all colors are much much darker and I would basically have to start from scratch again!

Before I do that I'd like to be sure that I have the right palette.

So is this one ("RealityC.pal") as close as I can get to the real colors of the NES, or do I have to look elsewhere?

The signal the NES outputs is flakey at best.  Certain "colors" actually can appear to be two distinctly different colors on a TV depending on the colors surrounding the pixel.  This is one of the causes for the many variations of palettes and whatnot.

No RGB palette will ever be dead-on.  And as far as accuracy goes -- pretty much all the mainstream palettes available are close enough to be considered "accurate", but again due to them being RGB they'll never be 100%.

If you want the closest thing you'll get to TV output, take a look at the NTSC filter in some emus (NEStopia uses it).  It does not use an RGB palette, and instead actually emulates the NTSC signal sent to the TV (even emulates dot crawl and other NTSC artifacts)