Florence: what the heck is . . .

You mean here

http://www.canottierifirenze.com/hcm/hcm30648-1-Home.html

Thanks, Christi. I'm not sure what you meant by "here". When I clicked on the word "here" in your message, I got a Google map with the round arrow pointing at "Societa Canottieri Firenze", which seems to be a rowing club. I don't think that's what I was seeing. What I was seeing is a long thin grassy area on the river side of the great big wall that runs all along the river; in other words, the grassy area below and to the river side of the Lungarno.

Try it now

it is the rowing club

The rowing club is housed past at a little green door just under the place where the corridor vasariano makes a turn direction Ponte Vecchio. The building goes underground and when the river is low they can access the grassy strip. They have a watertight door to the river, so that when the river goes up they simply take everything inside and close the door. The rowing club is a sort of historic institution in Florence - consider it as the Italian version of a top British golf club.

https://www.tripadvisor.com/Restaurant_Review-g187895-d4597115-Reviews-Canottieri_Firenze-Florence_Tuscany.html

Nice find lachera - looks like you can eat there!

Nice spot! Yelp reviews (in Italian only) seem to indicate that it's only open to members but (roughly translated)...

"Substantially open to members only, however, you can reserve a table, attend monthly theme nights or a party or an event by agreeing with the managers of the restaurant at the club."

So I guess it won't hurt to ask?

Review in Italian with photos. http://www.ioamofirenze.it/canottieri-firenze/
I suspect outsiders should be accompanied by a club member but you are right, you can always try to ask.