As a general rule most States do not have any insurance requirements for Handymen. You should however at the very least have Handyman Liability insurance. Liability insurance is your basic business insurance. If you need to pull a permit or register with a City or Town for a job they will in most instances require you provide them with a certificate of insurance showing you have general Liability coverage, they may also require you provide a surety bond.
You will need to check in your local area what the requirements are. They vary greatly from town to town.