My boss and major stakeholder (and CEO) in the business that I work as an engineer is pretty much busy all the time. In the USA where she is mainly based, she has an employee that holds the position of personal executive assistant to CEO and assists her in several personal situations as well. The company is pretty small (< 40 employees worldwide) and the CEO being pretty much busy, she many times asks us in the office in Greece to help her in personal situations, while it is not our job description (examples are taking her personal car to the car dealership). Actually, she does not ask for help as a favor, but she actually demands one of us (whoever is less busy in the actual work) to be available for her personal business task. How you would advise to handle situations like that without insulting her or even give an impression that I do not feel comfortable handing CEO's personal business ?
The fact she is the CEO of the company you work for and basically is giving you your paycheck doesn't give her the right to assign you to any job she wants. You probably signed for a specific job description and while some flexibility is normal it seems in your situation it goes way beyond a reasonable flexibility.
Although I'm very keen on helping people at work - were them colleagues or bosses- with matters not related to the work, I also reserve myself the right to set boundaries and/or refuse (or ignore) right away such requests that are not closely related to my job function. I already consider it a bit rude for her to ask knowing it will be harder for employees to refuse a CEO's request, but if it's such a recurent request, I find it totally unprofessionnal. If she needs somebody to handle her personnal stuff that bad, she should probably hire a personnal assistant whose role would be to handle personnal matters.
So to answer your question, I don't think that refusing such a request by saying "I'm sorry this is not even close to my job description so I won't help you" should be considered insulting. It definitely sends the message you're not comfortable handling her personnal requests, but if you were comfortable doing it you wouldn't be asking this question...
Now of course, as apparently this CEO has a tendency to mix up personnal and professionnal stuff together, the risk is great that you would be let go rather soon than later, was it for a false official reason to remain legally compliant. But it may not be that bad in comparison to basically, bits by bits, becoming this CEO's slave. Now it's her car, what will it be tomorrow ? Gardenning, painting, unclogging the toilets ?
If you can't afford to lose your job, I'm affraid there is no graceful way to refuse such requests on a long term basis, as you may find a good excuse once or twice, but it will soon become obvious you're just making up excuses at some point.
How you would advise to handle situations like that without insulting her?
It is too late to handle it.
You are in a small company, or at least a small company culture where it is expected that the employees coddle the CEO. This behavior has been permeated by the fact that when the CEO demands it, employees just do it. This is highly unlikely to change at this point. The best shot of altering the CEO's behavior was when the initial out of bounds request was made.
In short, you are going to have to suck it up and do it, or find another place of employment if it bothers you enough. The behavior won't change as the expectation has been set and met.