Hiring yourself in Missouri will not require worker's comp by law, but not having it will mean you won't get work from most companies. Homeowners seldom check for that.
Worker's comp is an insurance policy that protects the employer from lawsuits. Most employees think that it is for their benefit, but really, it is just an insurance policy to keep the employer from being sued by the employees. Hence, there is no value in insuring yourself against lawsuits by you. If purchased, however, it will pay your medical bills and could provide payments for lost work or permanent disability.
Most of the time, bigger companies will require that the insurance company send them a certificate of insurance, often naming them as "an additional insured". This protects them from being sued by the injured employees also.
Without that policy, you won't get their work.