IMHO unless you can get lots of customers then you won't make enough money to make the business viable, most web designers (myself included) do offer support / maintenance contracts for clients who need / request it, but it will depend on what you want from the following (or more).
- backups
- website edits
- hosting
Your best bet is to find someone who specialises in the software (if any) that your website works on, this will (hopefully) make sure you get someone who knows how to work with your site.
I have customers who do all or some of the above, some just want regular backups of the site done, some want some help making edits to the website (they pay for XX hours per week / month) or they just want to move their hosting.