I have the same issue, and ended up building myself a coffin. Soundproofing a rental apartment would have been an insane waste of money that I didn't have anyways, and sleeping with headphones on was impossible to me due to how painful it would get, having the plugs in when I slept on my side, pushing them further in.

During the day, I used the noise canceling earphones solutions most mentioned and that worked really well, but thank heavens I'm not claustrophobic.

If it gets so bad you need to do that, make sure you get good enough ventilation, an easy way to open it, and personally since I lived with roommates, I ended up also wiring an emergency alert button, so in case anything happened to me in there, they could come into my room and rescue me. One of them even made me a sign for "The corpse is out/in" so in case of emergency or a fire, they would go and wake me up so I could evacuate too. Plus two USB ports, one for a night light, and one to charge my cellphone.

This message is getting kind of long, so let me know if you want more info about it. But in a nutshell it was just an iron frame, plywood panels, for the outer and inner walls, and rock wool for the in-between of the walls. Two doors, one on the upper panel, one on the side, so I could just roll myself out if it in case of emergency. And most importantly, and to repeat myself, ventilation.

A small custom mattress, and lining on the walls to prevent splinters, and you're all set.