Ехать туда отдыхать в ближайшее время я бы не советовал.
Although I have no immediate plans, I think it would be very interesting to go there, particularly if it's possible to swim and get a tan at the same time.... I don't like packed beaches anyway.

I wonder what money they use, and how they support themselves? Particularly if there is no longer much tourism going on...

Regarding non-Russians/locals going there. Anyone wanting to try it would have to investigate beforehand.

I had an experience with this when I travelled through Pridnestrovie, I didn't have any information at all, apart from what some local people in Kishinev (capital of Moldova) told me. I had planned to simply take the train from Kishinev to Odessa, but they explained that Pridnestrovie was "in the way" and that the train was held up there for hours, in the middle of the night. Instead of train, they recommended marshrutka/bus. Because they were local, they didn't know that the bus was not allowed for foreigners (meaning non-ex-USSR people). I am sure the person who sold me the ticket could hear that I was not a local person, so they really should have mentioned it, or asked to check my passport. But they didn't.

When we got to the border, the bus driver got really irritated with me because I hadn't told him that I was a foreigner when I boarded the bus. (He never asked!) "Never do this again", he said angrily in front of the whole bus. But there was no actual problem, they let me through and issued a border visa. The other people on the bus didn't know about the restriction either.

After reading about Basil77's mother and her holiday it occurrered to me to check Youtube and I found some holiday clips by Russian speaking people. It doesn't seem like anyone else goes there.

You can really see the shadow of a glamorous past in this town, and the weather seems to be constantly sunny. The sea is a very pretty turquoise shade, but the beaches are not soft white sand but more gravel-like sand (this is a minus, in my opinion).

I also got the impression that Gagra can only be reached by car/bus (possibly also train). It makes sense - probably no airport in an "unofficial" country.
Hence: Land route via either Russia or Georgia.

Another thing on the plus side must be that it's cheap.

I feel sorry for the people. They absolutely don't want to be in Georgia, and Russia can/won't accept them so they are stuck in no-mans-land. I think it's a nice gesture of goodwill for anyone who feels inspired, to go there.

(On the other hand, as Basil77 suggested - Crimea might be a more fun option with more to see and do! I have never been there....)