You should at least have a general contract agreement with your handyman that has a hold harmless clause in it.In addition your handy man should carry his own liabilty and workmans comp insurance insurance as an independant contractor.Probably would not hurt to talk to an attorney about minimum safe guards for you association.

It all depends on California law and requirements, and your association documents.

Liability insurance to require of contractors is a real regional question, meaning there could be multiple answers within the same state. 
 
One would hope that your HOA has an insurance agent that provides insurance coverage for your building structure, common areas, and HOA Board. I would ask your insurance agent, if they don't know the answer it would be time to switch insurance agents.

Be careful. Our assoc. had a handyman who supposedly had insurance in the past that no one questioned. This year we asked for a certificate of insurance and got one. But.... checked with ins agent on certificate and agent said he never heard of the guy. Called the insurance company also shown on form and they never heard of him either. Went on internet and found that there are actually web sites that show you how to fake a certificate!! Beware!

regardless of state laws, it is recommended that associations carry workmens comp in addition to general liability. Handymen are notorious for not carrying their own insurance because it can be extremely expensive depending on the state.

Susan is right. Our HOA purchased a workers comp policy a few years ago even though we have no employees. The policy covers us in the event of an accident or injury to an uninsured worker on our property.

Consult with your insurance agent/broker and then call your attorney to make sure your coverage are right. We carry workers comp for our association even though we have no employees just in case a vendor's policy lapsed. The cost is minimal for workers comp in our policy for the association.

I agree with everyone here.  
It is important to check that each service provider on your property is insured. It is also important that your association carries insurance to cover workmen accidents.  
Your Association's insurance company can review your current coverage and make suggestions for how much is suggested.  
 
Similarly, every unit owner needs to carry individual condo insurance. Such insurance should cover liability in case someone hurts themselves inside your home.  
 
In fact, my mortgage company demands such insurance, and demands a certain coverage...and demands to see my condo insurance and the Association's insurance certificate every single year.  
 
Even if your mortgage company does not demand this..do not for a minute think you can get away with not carrying this, and save on this cost.  
 
The first time someone slips and falls and dies on your property and their family sues your Association...your Association will lose, will go bankrupt, and every unit owner as well will go bankrupt as you are strapped with a 15 million dollar settlement. Remember, we live in America, the home of the-McDonalds-hot -coffee-sue-happy everyone-trying-to-get-rich-quick-land-of-freedom.  
 
LOOK INTO THIS NOW. Make sure you are covered and your Association is covered.  
 
It is not sufficient for you to assume you are covered, and hope someone else is taking care of this