We'll probably never see an official Half Life 3 release from Valve and at this point, most seem OK with it. Valve is the company that makes standalone mods of its own games now, rather than new titles. That's a shame, but it's been more than 12 years since Valve released the last "Episode" in its Half Life series and any new game wouldn't be anywhere near as groundbreaking as the series once was.
But if you do really want to play Half Life 3, a collection of mod developers are working on something that will scratch that itch. Their own, standalone HL3.
Called Project Borealis, the game is based on former Valve Half Life writer, Marc Laidlow's synopsis of the original plan for Half Life 3 that Valve never realized. It will feature head crabs, physics guns, and a conclusion to the long-standing Half Life story. And it's coming along nicely.
The latest two update videos showcase a game that's fully under development, with some basic level layouts, some concept designs for more expansive maps, and some new gameplay mechanics, like heavily destructible environments. We also see how the level creation process works, starting at basic concepts, moving to blocky samples, before additional concepts and new refinements are made.
Most of it is still using placeholder elements, but it's looking good. You can follow more on the game's development on its official site.