Are my hairline cracked or broken carbon fibre handlebars broken or useable? Can they be easily repaired?

If the main body off The handlebar is not cracked, then these will repair. Look up `carbon fishing pole repair` My mate mends fishing poles. They never break in the same place. He wraps new fabric around The breaks and heat cures the section.

Personally..... I'd bin them and replace....as it's a used bike and this is from it, then as you are probably not aware of the history of the bike...  Just my opinion as a Bike Mechanic...and no not flogging new parts...my point of view. I buy in bikes for customers requirements and to be honest if I found this during the usual strip & rebuild I do, I'd swap it out.

Your call and You pays your money and takes your choice

How much do you value your face?

Cos that's what you'll land on if they break.

FSA K-Wings ? Mine did that (tho not as bad) and have been fine for years....not got worse etc. They're apparently known for doing this and no it's not structural so fine to keep using them 

I want the structure that connects me to my bike and brakes to be beyond question.  As these are second hand you have no way of knowing how they got in that state (and they look pretty bad)- treat yourself to a new pair. 

You've asked the same on bike radar. Whats the answer you want to hear?

Personally they look to be in a state to me and I'd replace them with new alu ones that I would trust.

Looks to be just the cable channels breaking away from the main body of the bars

Seen this before on a set of Fsa plasma bars

If your worried then id get a deremel and remove the channels altogether and check the actual bar, if all ok then carry on using them, if they have cracked into the main handle bar then bin em and buy new ones