Sony owns the market for consoles, being installed in something like 30 million homes? The GBA (and all the ones preceding it) has a similar installation base - I think one of the stats was "it sold a system every fifteen minutes since the early 90's"? It was in reference to a milestone for the gameboy, several months ago.
That said, it would take a monumental effort to overcome not only Sony but MS and Nintendo to have any sort of presence in the market, AND produce their own titles because 3rd party developers wouldn't latch onto the system. Why? One failure after another - Game Gear, Sega CD, 32X, Saturn, and the DC (Yes, the DC failed- by being put to pasture it means it 'failed', as opposed to 'not failing' which would entail not being put to pasture, you see). For Sega to have any chance of overcoming Sony they'd have to have a large library available (because the next Sega system would not only have to deal w/ a PS3 but the PS2 as well - just like the DC vs. PS2 + PSOne) and a lot of third party support. The DC had interesting (and fun) games, but it didn't have 'popular' games (like Vice City, or Tekken, etc).
And the handheld market - nevermind. If there was ever a monopoly in the handheld market, nintendo has it. And not because of underhanded tactics (not that I know of) but their portables are simply awesome. Yes, I know the GBA and it's backlight (or lack thereof) issues, but it's still one fantastic system. And I don't care that they're simply porting SNES games - there's a difference between sitting in front of one's computer/tv and sitting in a train/plane/boat/wherever and playing these games.
Sega isn't MS - it doesn't have the money to go through a protracted battle with three other companies. They've already lost twice to Sony, how is it remotely feasible that Sega would do battle again? Not to mention there are two other systems already available. It's a pipe dream. Sega might be for the fans, but their fans are gradually dwindling, not to mention these are the same fans who pirated the DC out of existence.