Lots of good ideas, but I kind of skimmed so if I repeat I apologize. :)
Divide and conquer. If possible, sign up one or two kids at a time for a summer day camp. Around here the camps are pretty expensive, but we usually manage to get the kids into one or two camps over the course of the summer. They aren’t all gone at the same time, so they get to take turns getting some one-on-one time with me. My 12 year old daughter really appreciates this break from her younger brothers.
Be strict with your expectations and model what you want them to do when they disagree. Don’t step in at the first sign of trouble, but encourage them to work it out themselves. I draw the line at mean names and physical fighting but give positive reinforcement when they come up with a solution themselves. There’s a book called Siblings Without Rivalry that you may want to read. It has great tips for getting the kids to work together more.
Take the oldest child aside and encourage him or her to sometimes be a leader for the younger kids. My younger ones look up to their big sister so much and will resort to teasing her if they want her attention. I’ve straight out told her that the boys will do anything she wants if she’s nice to them. She will invent elaborate games for them to play and they are just thrilled to be doing something with her. Once in a while I will bribe her to keep the boys occupied!
Make them go outside. Stock up on inexpensive outside toys like a garden sprinkler, balls, hula hoops, bubbles, sidewalk chalk, or whatever. Bring out something different when the kids get bored.
Start a reward program. The kids earn checks (or stickers or tally marks) by getting along and doing chores. They lose checks for fighting or not listening. When the kids have earned a total of, say, 30 checks, they can choose a special activity like getting ice cream or going to a favorite place. Since earning the reward depends on all of them keeping the checks they earn, they will actually encourage each other to be nice, listen, help, and avoid fighting.
Invite friends over. It may seem strange to think that having more kids around will be easier, but when my kids have friends over they are so busy they forget to bother each other.
Go places. Arrange to meet the kids’ friends at a park or something. Find the free or no cost entertainment options in your area and take advantage of them. Like having friends over, doing something interesting out in public gives the kids something to do besides think of ways to annoy each other. Get out of the house and stay busy. Don’t use the TV or video games too much (reserve them for times you REALLY need a bit of peace and quiet).
There are plenty of resources online with ideas for activities and games for kids. Look some up!
By the way, it is OK to insist your kids give you a break. We have a summer routine that has a mommy break worked in. When I have my afternoon coffee, no one is allowed to talk to me unless there’s an emergency. I sit in the shade with my coffee and a book, sip slowly, and keep an eye on the kids while they play. When I’m done, they can ask me to push swings or catch bugs or whatever. The break is nice for me and I think it’s important for kids to be able to respect a parent’s needs as well. It isn’t always all about them!