No broadswords, no fancy vests with sleeping gas grenades. Sinews against thinews, whatever they are and however you spell them.
My money is on the Giant Cimmerian, he’s gone hand to hand with monsters, while the Man of Bronze dresses up like an alligator and goes mostly after cheap crooks.
Yeh, but Doc Savage has all the advantage in brains, and does daily exercises to keep his body fit and his mind sharp.
Besides, going by the 1975 movie, he’s a master of every known martial art from bo jitsu to fisticuffs!
I believe the correct terms are “sinews” and “thews”.
My money is on Conan. The good doctor always made an effort to take his opponents alive and mostly unharmed. That will make him hesitant when he should be aggressive.
I’m pretty sure that in one of the books Conan defeats a Giant Gorilla in hand-to-hand combat.
After that Doc Savage will be a piece of cake. 
Doc Savage, if he doesn’t hold anything back. He took down a polar bear with his bare hands.
Doc will win. Conan will go thundering “I’m Conan the Barbarian, I yam, I yam,” and Doc will tell him his shoelaces are untied. Conan, will look down, notice he’s not wearing shoes, and wham.
Brains (plus some brawn) over pure brawn every time.
Wait, since when is Conan supposed to be a dummy? Sure, he might not have been a Rhodes Scholar, but he was the Cimmerian equivelant. He’s well-educated and widely travelled, and if he isn’t a wizard he’s probably learned more about the Black Arts (from all the times they’ve been used against him) and is well-prepared for a huge array of attacks, mental and physical. Conan has commanded bandits and pirates in battle, led groups of thieves in cunning raids, and generally gone it all.
Yeah, Doc Savage exercises and studies science. Conan has been involved with more fights that Doc Savage could imagine, against every kind of opponent there is. Plus, he’s frickin’ Conan. He’s like a barbarian Captain America. Yes, Doc Savage is good. But this is like putting a skilled kung-fu fighter in a cage match against one of the MMA’s best. Conan can fight fast, or brutal, or conservative, and he can box, or grapple - any way you need.
They’re twins. The two men would size each other up, wrestle to a draw, then go whoring together.
Brainglutton’s link does sort of look like Arnold.
I am a fan of both, & find it too close to call.
I suspect it would boil down to luck.
Then Conan wins because he has the Luck-Charm of Zorgoth or some such thing Doc Savage would be too scientific-skeptical to pick up off the ground.
I bet on the Man Of Bronze. His superior knowledge of science will see him through.
Also, us Docs have to stick together.
One thing is Doc Savage always has concealed items in his body, even if you strip him naked. As for strength and cunning I’d say their both about the same. Though Conan will get pass out drunk, while Doc Savage maintains a strict dietary regime which doesn’t include getting drunk.
I recall a filksong called “Thews,” to the tune of “Men of Harlech.” It begins:
Why be scared of evil wizards,
Fire-breathing scaly lizards,
Cannibals who’ll eat your gizzards?
All you need are thews!
Never mind that old book-learning
Fencing, dodging, weaving, turning
When with strength your thews are burning
Nought stands in your way!
Thews on chest and shoulders
Thews as big as boulders
Hairy legs
Like brandy kegs
Will flatten foes and fiends like filing folders!
Baddies groaning
Princesses moaning
Chastity and kingdoms both dethroning!
Then next day your agent’s phoning
On the road again!
Dr. Clark Savage, Jr. The Man of Bronze.
Did Conan ever punch out a shark? huh?
In unarmed combat, Conan wins, he’s just got too much native ability, brawn and experience at it, even for Doc Savage. Anything involving figuring things out, Doc Savage wins, though Conan’s skills in this area are not inconsiderable. But Doc Savage’s skills in this area are far past Conan’s.
Doc Savage. He’ll be better armed, and he’ll be pragmatic enough to say, "Hmm. this guy is obviously much larger, stronger, and skilled in melee fighting. Best to cheat here.
The OP rules out weapons. Unarmed combat only.
Arnold decked a camel during the filming of Conan the Barbarian. I guess that doesn’t count…
I burning your mercy bullets!