I did, in fact, repair an heirloom yo-yo quilt that was full-size and had been put on a shelf for years because it was coming apart. A very aged widower contacted me as his late wife had made it as he treasured it, but wanted to use it. Because I did not want to compromise the hand work with machine "tacking" to a back, I did it by hand but made sure NOT to use a sheet. I only used a sheet once for handwork and almost crippled myself. Sheets may make fine backings for machine work; not hand.
Interesting that it was a color very similar to the one you show and indeed, I did back it with a very pale blue backing, but one that had a muted tone-on-tone to give it some interest other than a complete solid.
When I attached the yo-yos, I took a very small stitch at the beginning and ending of seam lines of the yo-yos, using a very fine, neutral, lighter color thread. I then passed the needle and thread between the top and bottom layer of one of the yo-yos to the next end of the seam and took another very small stitch over the seam and through the backing, etc., for the entire quilt. Of course, I repaired seams between the yo-yos as needed as I came to them.
So as not to give the appearance of binding on the top, I simply left about a 2" border on the backing and turned it back on itself twice = a 1" "hem." I mitered the corners and finished it by hand as well. He was very, very happy with the end result and handsomely rewarded my considerable effort. It was worth it to just see his reaction amid his sadness. And the quilt was now able to be handled and gently laundered without further damage.
Incidentally, I recently saw a YouTube by SewVeryEasy with very much the same procedure. I think she called it a "rolled" hem with mitered corners. It always looks so nice. (I found it. Go to:
https://www.youtube.com/watch?v=DVrGV2xSjbk&t=10s )
It is a lovely thing you do. Hope this helps. Quilting is all about the heart involved in it.