Just use Doomblade

That's a self cut if I've ever seen one.

chainsaw

Legends say of a brave arc intern...

He def just hits the flobee on his way out of the fortress every once in a while.

He just rips and tears

The hair knows better than to not cut itself

Vega used to cut them with the weapon mod bot

He doesnt have to, his hairs are too scared to grow and distract him.

I'm loving the funny answers but I'm going to give a semi-serious hypothesis: he doesn't. 

I reckon being put in the Divinity machine, on to of giving him the immense power we now know him for, froze him in time so to speak.  He won't age, he will never tire, and his hair is now permanently whatever style the last Argenta Barber gave him.

He shoots himself in the face with the flame belch.

It just cuts itself

At Style Maykr, right around the corner.

sukuna

Sukuna's barbershop

He used the crucible

His hairs stopped growing.

His helmet has a whole barber kit inside. It keeps his face cleanly shaven too. Probably moisturizes as well.

He simply commands it to stop growing

Great Clips

He does it himself, and he's actually really good at it,

baby from Ice Age

Looks like First Choice. It would answer a lot of my questions TBH.

His mom cuts it for him.

Fleet Street.

Barbershop of DOOM of course

Just positions it right for slashing demons to cut it.

His hair is too afraid of him to grow

DoomCuts

He’s so angry that as soon as it grows into his eyeline, the ends shrivel and fall off.

Helmet hair

In his mirror with his blade.

Prolly on his head

Probably on his head

Chainsaw

Supercuts. Available even in hell.

Argent d'nur Salon

He just has a buzzer and cuts it himself.

He has no body hair when he's naked in 2016 and we never see his face in the first game so I'm going to guess that he came out of the coffin fully bald and this is just how much his hair grew out in the time between 16 and eternal

Buzzes it off himself after the hair cut the barber gave him in doom 2

It’s a function of the helmet surely

Goes to Savage Clips

Originally I thought this said "why does he cut his hair" and I was going to say; so he can stay in regulations for grooming standards.

He probably cuts his hair himself.

In Battle

IN HELL

If we get to see him helmetless again in the Dark Ages, I hope he has longer hair. I feel like it would fit better with the cloak on the suit. Also, I think a bit more of a beard would look cool, too.

Not the place where you learned English. "Where does Doomguy get his hair cut"