The quick answer is no. An emphatic no, as a matter of fact. Every business should and can have a website. No matter how small.
In a recent survey of over 4,000 businesses with 1-5 employees 60% didn't have a website. Of those 60% only 12% had a Facebook business page.
Nowadays if a person can't find your website you are pretty much invisible. Why would anyone do this to their business when they're working so hard to succeed?
Small business owners don't have the time to figure out how to make a website, no matter how simple it is. They simply have too much to do. And as for cost, I blame the scamming sales calls that businesses get for making people think they need a fancy website or that they need to pay money to be listed on Google or that a website is going to cost at least $1,000 dollars or more.
You can have an easy to use and edit website for a reasonable price. Please don't believe the callers who tell you you need to pay to get listed on Google. If you have a website that is filled out correctly and has relevant information for your customers, Google will find you and show your site in search results.
Find a local business who works with small to micro businesses or let Plain Language Marketing make you an easy to manage website. No need to call on your "web person" to make small changes. Give yourself the power to have a strong online presence and watch your business grow.