You've not said what size your monitor is, so your question is kind of hard to answer. However, I have a 22" LCD monitor with a native resolution of 1680x1050 and an apect ration of 6:10. As a test I was able to set Half-Life 2's resolution at 1024x768 and play the game in full screen (without black bars anywhere). I myself play the game at my native resolution (1680x1050) and have never had a problem.
Bottom line is you can play at that resolution in full screen without issue provided your monitor supports that resolution. If you're having issues, you might want to check your video drivers
as this game plays in full screen whatever the resolution.
Also, in the in-game settings under video, make sure you set the "Aspect Ratio" to 4:3 if you're going to play at 1024x768 - that might be your problem.
If you continue to have issues with the game you may need to reinstall steam.
Hope this helps.