OP is talking about installing the game, not running it. It's a documented issue that some older installers will not run properly without the 3.5 framework active.
They stopped enabling 3.5 on new installs a while back--without it you loose older framework support. Forget exactly when this happened, may have gone as far back as Vista or 7... but it wasn't active in my 8.0 fresh install to my old XP PC, nor on my ASUS laptop (refurbed older model, but loaded with 8.0). A store bought system may have it enabled to support older bundled software, but a fresh install from a MS DVD won't. This was a big problem for people with XI and other older games as well... both 3.5 and DirectPlay had to be toggled for that game to work, so some of you guys may already have them turned on and never ran up against it with XIV, but it could potentially be an issue here as well.
Keep in mind that XIV was initially developed with the intention of running it on XP (even 2.0 ARR was targeted to run on XP initially)... which means there is a lot of legacy stuff in there. As in MSIE8, .Net 3.5, and DirectX9. So this could very well be an issue with fresh 8.1 installations.
@OP, you may also want to enable that legacy support for DirectPlay as well (old DX9 feature that may not be available by default as well). That should also be available in the list for turning Windows Features on or off, under the Legacy section.
Edit:
Just out of curiosity, I poked around in the installer with TextPad and there appears to be a .Net Framework check going on. Error messages are there in plain text about failing version check in mscoree.dll because of deprecation in the 4.0 framework, and further down another message in plain text about .Net framework 1.0-3.5 not found. Whether that is the root cause of the OP's problem or not remains to be seen. But it does appear that the installer may be checking for .Net and it may throw up on itself if it doesn't find the legacy components available. There also appears to be a check for DirectPlay and at least a specific version level on the DirectX files. So there may still be some weirdness where people need to enable legacy support or run the old DX runtimes again like we had to do in the early days of Vista/7.