Relax, hell! We're rolling!
In the history of college sports, what you look for has never been done.
In fact, I couldn't find a single instance of any school winning national titles in three of the Big Four sports (let alone all four) in a single year.
I know two have been won in a single year, most recently (I think) in 2014 by UConn when both their men and women won the NCAA basketball championships. But I couldn't find any evidence that any school ever won three of the four in one year.
[though Stanford has had some crazy-good runs in combinations of several of the Olympic sports]
Heck, only 29 universities have EVER (forget in a single year) won titles in more than one of the Big Four. Tennessee is one of those, since we have national titles in both football and women's basketball. Nine programs have won national titles in three of the four,* and one of those can claim a national crown in all four at some point in their history. Yep, Stanford. Though some folks think Stanford's football title claims from back before WWII are pretty iffy (but the NCAA recognizes them).
So I'd love to see our BasketVols or BaseVols -- or both! -- win a title this year to make us the tenth university ever to win it all in three of the Big Four sports. That'd be cool.
And I'm with the OP: winning all four in a single year would be just simply awesome. We can dream.
We're rolling!
Go Vols!
* The nine are Cal, Florida, Maryland, Michigan, Ohio State, Stanford, Texas, UCLA, and USC.
