My wife & I were watching a "Highway to Heaven" episode with our grands last week and in one scene, the character went to the Post Office and asked for his mail care of "General Delivery". The kids had no idea what that was and after explaining it to them, we all got to wondering whether that service is still available any more. I will be asking the next time I go to my favorite branch, but maybe you all know... Does the USPS still offer that service?

yes, some post offices still do it, but it is only for very temporary situations. 

Yes, I've seen people pick up mail at my local post office using that - either  people who are vacationing at a local park, or hikers passing through on the Pacific Trail.

I called last year and asked about it.  Requires a specific procedure/request in advance (like a change of address) you can't just tell someone to mail your stuff to General Delivery xyz and expect to receive it.  If they don't have the proper paperwork, the receiving PO will probably just return the mail to sender.

That is very common for military members to use when they get transferred and don't have a home for the first month when they arrive to a new area.  The gov't pays for a hotel for a certain number of days so it is typical for military to get their mail sent general delivery til they can open a post office box or have a real home address to provide.