Is it possible that a billboard Gui will stay in the same size?
It already does. If you are looking for it to stay the same size in one location, then you have to use a SurfaceGui.
To start this off, there are 2 things you need to know about, Scale and Offset. Scale is a percentage of what the gui will take up on your screen and offset is the amount of pixels it will take up. When you open your properties tab you will see size it will look similar to this {XScale,XOffset,YScale,YOffset}, The reason your billboard gui isn't retaining the same size is because you're using scale instead of offset if you want the billboard gui to always be the same size use offset. Hope this helps!
Go into your billboardgui and whatever it's inside it, then go to their size, press the down arrow for size, then change scale to 1 and offset to 0 for both then adjust size accordingly