I have all my existing sheets set up for about half a page.
I fancy you may have your terms a little confused here. By "sheets" do you mean spreadsheet documents or individual sheets within those documents? "Page" refers to a printed page, so you seem to be saying here that your spreadsheets only ever cover half a printed page or that every page of a multi-page document is half empty.
I can get two on a single display if I need to do that, but mostly I want to see the data I am about to enter.
Now you are talking about "display" instead. By "page" did you perhaps mean "window"?
However, every once in a while I will open one to full screen and save it or I will create a new one from the top tier of LibreOffice and this screws up every one of my carefully sized sheets. Is there a way to lock in the desired size as a default? I've checked preferences and I don't see anything.
A product will usually open with its window at the same size and position as when it was last closed, won't it? So it's not that you have "all your existing sheets" set up this way, but that since you usually use LibreOffice that way, you will generally have closed it with a half-screen window (if that's indeed what you mean) and that's the way it will open - not how it was the last time you opened the particular current spreadsheet document.
I suspect that this is really a question for your operating system (which you have kept a secret) rather than LibreOffice as an application. I don't know if any operating system allows you to override the default behaviour. As a workaround, I imagine most operating systems allow you to tile windows either horizontally or vertically, and if you have two windows open, this may achieve exactly what you need on the fly and without any particular care being taken.
I trust this helps.
Brian Barker