Hi AntoineAtl,
Yes, the SharePoint Online service can host the user’s website. However, here are two situations that I would like to indicate:
1. The user has the existing website and wants to move it to Office 365.
I regret to inform you that it is unavailable to import an existing website into Office 365, so existing websites may need to be rebuild or modified to be hosted in Office 365.
2. The user wants the website to be hosted in Office 365 with the custom domain.
If this is the case, the user need to add the custom domain to Office365 first, and then set up the public website with the custom domain. Here are the links for your reference:
For Office 365 P plan, please refer to http://onlinehelp.microsoft.com/en-us/office365-smallbusinesses/gg549209.aspx.
For Office 365 E plan, please refer to http://onlinehelp.microsoft.com/en-us/office365-enterprises/gg570301.aspx.
Thanks,
Eben Liu