Whenever I try to save, it says it is unable.
Any solutions?

Found this regarding FO 2:
"Try this, uninstall, delete the program files interplay folder and check the registry for entries about Fallout 2 and delete that also then install."
I didn't find any obvious topic regarding FO and saving, is it FO2 you're having the problem with?

This is for Fallout 1.

I know that this may sound stupid and all but is your SAVEGAME folder writable by you or by the group you belong to?
If not, try and change the permissions so that you can write to that folder. Also, try running the game under the Administrator account (not one with administrative privileges, but the main admin one) and see if that helps.

I presently have no prior savegames and thus no SAVEGAME folder.
I am the only account (AKA the administrator).

Can you manually create the SAVEGAME folder there?
What OS are you running on?

I am running on Vista, and I tried creating a SAVEGAME folder in both the Fallout folder and the DATA folder (located in the Fallout folder).

I'd try a reinstall personally, it won't matter much either since you can't save. :p
The game's old and can cause some problems by being just that, after installing I'd also get a newer patch and install that before trying the game again. (it'll corrupt your savegames if you add it later)
1.3.4 is what's recommended by NMA and it's available here:
1.3.4

I have already reinstalled it, and even after installing the 1.2 and 1.34 patches, it did not work.
EDIT: Well, I chose to run it as an administrator, and now I can save. I suppose I was incorrect about being the administrator (I assumed that since I was the only account and thus the main account, I would have administrator privileges, so apologies to AndrewC).

No problem, most people confuse having administrator privileges with being on the Administrator account (Vista especially has some differences because of certain groups the main admin is part of - at least in Business and I presume Ultimate versions).
To stop having this problem you might try and install the game in a non-system directory (as in not in Program Files, Documents etc.) while under YOUR account (not the main admin one). Also, I would advise disabling UAC.