My DS was like this. Firstly, know that it will get better. It WILL get better. There probably won't be a magic day when you suddenly get 8hrs kip but gradually. I know that when you're in the middle of this, it feels as if it might never end and that you need to find the solution which will magic it all away. Sadly it's not like that. When you have a DC like this, it's as much about your practical 'survival'.
So here's what helped me:
(1) sharing night times with DH. DS could take hours to settle so we got into a pattern of I fed, DH settled. It meant we were having broken sleep, but sleep nevertheless.
(2) Co-sleeping. We didn't do it full- time (I can't sleep properly) but it did help to at least allow me to feel rested. Frankly it was a god-send!
(3) A sling! Oh how brilliant was my wrap sling. My DD spent much of her early life in it. It settles them, they sleep, you can eat using two hands, what's not to like.
(4) Accepting that it's ok not to put your DC down. If you spend all day with a feeding sleeping baby lying in your arms, the that's ok.
(5) Whem you need sleep, you need sleep, and if that means someone taking your DC somewhere else so that you can get three/four hours kip then so be it. If they cry, they cry. They'll be ok!
(6) BFing. 'Your baby is comfort sucking / using you as a dummy'. Damn right he was - the alternative was a screaming colicky non-sleeping nightmare.
(7) A dummy. Worked for DS but not all babies are interested of course (DD wasn't).
But first things first, you need some sleep. Who can hold the baby for you tonight / tomorrow so that you can pull yourself back from 'dangerously tired' to 'gobsmacking tired'?