Has there ever been a score in a tennis match like
6-0 0-6 6-0 0-6 6-0
If not, what's the closest to that score in grand slams/major tournaments?
To answer this, I would refer you to a message thread over at the men's tennis forums site where somebody did the research on these kinds of scorelines since 1980. See here.
On the women's side of tennis, I'm sure this has happened many times where the winner won 6-0, 6-0. You tend to see those kinds of scorelines once or twice for each grand slam tournament where a player like Serena or Sharapova beats a low-ranked opponent 6-0, 6-0. I don't think I've ever heard of a match where the loser recorded a 6-0 set against the winner though (where the winner won their two sets by a score of 6-0 also) - that would be very rare. I will research it though and post an edit in here if I can find a record of it ever happening.
Here is another good read about the types of scorelines you're asking about - which mentions that it's a more common score in the women's game.
EDIT: Ok - I found only two matches so far that fit your criteria (both on the womens side). If you look at the match results for this player (Ka-Po Tong) from 2000 - scroll down to the Shenzhen tournament and the SemiFinal match. She lost the match 6-0 0-6 6-0. Then in Fed Cup Qualifying rounds in 2003 - Maria Fernanda Alvarez (BOL) d.Zara Desilva (BER) 0-6 6-0 6-0 . You can find that result here.
For men, I could only find:
25 0-6 6-0 turnarounds (basically at least one big swing in the match, such that the loser of that 0-6 set storms back to win the match)
14 6-0 0-6 turnarounds (at least two big swings, where someone wins big, then loses big, then wins the match)
Similarly, in women's, I only found:
27 0-6 6-0 turnarounds
13 6-0 0-6 turnarounds
In terms of complete matches, I also came across the same 2003 Fed Cup Qualifying match between Alvarez and Desilva that jamauss found. I don't believe his other match was a WTA tournament.
I found one additional match, from the 1993 Fed Cup qualifiers, between Gabriela Beleni (UKR) and Vita Kaminskiate (LTU), which is verified here. It is the apparently even more unlikely 6-0 0-6 6-0.
I found no men's matches with all sets to nil with such shifts. I did not even find any runs of three consecutive to-nil sets with such criteria.
So with only about 1 of these simpler winning to losing to nil turnabouts per year on each tour, even with hundreds to thousands of matches from the database each year, it seems apparently very unlikely that a 5 setter will go as such in our lifetimes.
Then again, apparently statistical absurdities do happen every once in a while.