That's a neat idea Henry, what kind of base is that? I thought a base shows up on the map and everyone can attack it, although in my travels I never came up against one.
If private bases can be kept well, hidden, what would be a good cheap option for a noob like me and where is a decent location? Close to a mine and adv workbench, or close to ruins?
Bases are private like tents. You have a couple of options as well, I suggest checking Wiki for specifics.
Easiest way to do it is to buy a "Construction Site" for 10k caps (several places to do this, I do it in NCR), and take the construction worker to the location you want to build it. Tell him "Right here", and you'll get a construction site. Gather the materials, tell the guy to build it.
It's 100% secure, and you can buy as many as you'd like under the same faction name. You'll choose your faction name from a list when you purchase the first base. Use ~ffn command in client to get a list of available names.
The location is up to you. Henry and I do it differently for sure, but our way both works for our specific needs. Figure out what works best for you.