Oh, mama, sorry; that sounds so frustrating!
My 21 month old DD nurses every two hours round the clock, but not every five minutes. Is this a recent thing? If so then it's probably a growth spurt. Have you tried offering him solids and water first when he asks to nurse? I started that when DD was a year and it took a while--she had NO interest in solids at that age either--but she was hungry, and eventually as I kept offering, she figured out that solid food could actually satisfy hunger too.

If I were you, I would keep trying lots of different solids--in the hope of finding something he likes--and offering that first every time he wants to nurse. It sounds to me like he's hungry but hasn't realized that solid food can satisfy hunger. What solids do you give him? Meat was one of the first things my DD started to enjoy.
Oh, and my DD would NOT drink water for a long time. I started giving her a little bit of juice very watered down so it had some flavor, just to get her to drink it. She would drink that. Once she got used to drinking from a cup, I switched back to water (and I still called it "juice"

). Now she drinks plenty of water, and she'll sometimes even accept water as a substitute for milk.
Also, is he with you all the time? If so, can you have someone else watch him for a couple of hours? Sometimes they're more willing to eat solid food if it's offered by someone who isn't mama. Maybe if someone else can get him to eat a solid meal, then he won't be so hungry and will give you more of a break.
Oh, but if this is a really recent thing, it could be illness or teething, especially since he's fussy. If he's fighting an illness, then this could be his body trying to get the antibodies it needs (from your milk) to help him fight the infection. In which case the best thing to do is just let him have as much as he wants until he fights it off. Something to consider...