How can I play Half Life 1 (or any GoldSrc game) in a borderless window?

You can set the game to run in a borderless window through Steam:

Right-click the game in your library, go to 'Properties...', select 'Set Launch Options', type in -windowed -noborder, and click 'OK'. Twice.

This should logically work for all Goldsource games.

For all games that can run in windowed mode (including Half Life 1), you can also consider using fullscreenizer, an open source tool that forces a chosen window to become fullscreen borderless.

This is especially useful for games that do support windowed mode but don't support fullscreen borderless. But as Joachim pointed out, this specific game does. I personally use it for certain indie titles.