If you know for sure exactly where your game will be on screen all the time, then you can setup a scene with a static image to frame the game quite easily.
So I'm assuming you have a game / stream that takes up some fraction of your scene. The rest of the space will be black. You can drag the source to any part of the stream by editing the scene. Then make an image and cut out the bit where the source would be. So you can take a screenshot or something, cut out where the stream would be and make it a separate scene.
There's a few guides on the web that show this with screenshots etc. Just google something like "overlay OBS scene" or similar and take a look. Hopefully this was what you were aiming for, I'm at work so would post links but most are blocked.
