I sympathise - I had two DCs very close together, anc the year after DC2 was born was probably the hardest of my life. It did eventually get easier. Maybe when he was about 1? I can't remember exactly. But the combination of baby and toddler was exhausting.
As a previous poster has said, it sounds like sleep is the big issue. However, at 5 months their sleep is still undergoing big changes - it could/probably will all change on its own in a few weeks or months. But you might be able to help your baby sleep better by trying to get him in a bit of a schedule during the day, giving him a 'dream feed' around 11ish maybe, or straight up sleep training (which usually means some form of leaving him to cry).
Also work on getting good naps in during the day, even if it means taking him for a walk to get him to sleep. I'd expect him to need a nap after being awake for about 2 hours at that age. Maybe a bit more or a bit less. Good daytime sleep often helps with night time sleep.
I have a 6 month old who currently has about 3 naps during the day, goes down for the night between 6-7 pm, wakes for a feed around 12.30 and then 3.30, and then is usually up for the day around 4.45 am! I feel like I barely sleep after 12.30. Not ideal, but I can get a few hours in if I go to bed early.
Re constantly wanting to be held, is he happy if sat up with some cushions with a toy (with you in view), or with you talking to him? If you can get him to start doing that for a few minutes at a time, he might start getting more comfortable being out of your arms. You could gradually extend the time he is sat down. Does he have a playmat or a bouncer?