Some news (or lack of news) from Gabe Newell about Valve's franchises.
This comment from Newell is particularly revealing:
The issue with Half-Life for me is that I was involved in a much higher percentage of the decisions about the games, so it's hard for me to look at them as anything other than a series of things I regret. There's no information in my response about what we'll do in the future. It's simply easier for me to be a fan of things that in which I was less directive.
He also says that there's a chance that Valve will introduce a new IP that takes place in the Half-Life/Portal universe. Of course that's not a confirmation, but if the story is continued, perhaps we'll see it told through a new IP rather than a direct Half-Life 2 sequel.
There's also this news from a year ago, where the sole writer of Half-Life 1 and 2 announced his retirement from Valve. So if there is a new Half-Life game, it would likely be written by a new writer (or team of writers). Whatever story Half-Life's writer had in mind may or may not be continued.
My guess from all of this - and it really is a guess - is that Half-Life 3 is not in active development. Valve is a company that explores new approaches to gaming, so if there is a continuation to the story it would need to be break new ground as an FPS or explore some other innovative mechanics. Because the writing does not seem to be a priority here, Valve are possibly waiting on the right gameplay mechanics to emerge before making a continuation they deem worthy.
Now here's a question for you all: do you think Half-Life 2 needs a sequel, or could the story be continued as a new IP?