Taking your files to the cloud is a great way to store files that you use in your organization. You can store files in a SharePoint team site, or in your One Drive for Business.
Save team files to a SharePoint site:
Go to your Office 365 team site. On the Office 365 Home page, choose Sites > Team Site.
On the team site go to Documents and drag files to the documents library.
Save your files to OneDrive for Business:
Choose , and then choose OneDrive.
Choose Upload, browse to files and then choose Open.
When you store your files in team sites or in One Drive for Business, you and people you collaborate with can access these files from anywhere with an internet connection. Read full article.
By default you get 1TB per site collection, and you can have up to 500,000 site collection and up to 2000 sub sites per collection, depending on your subscription. For details see SharePoint Online Limits. You can also purchase more storage.
For One Drive for Business, by default you get 1TB of storage. You can also buy more storage if eligible. See Set your OneDrive for Business storage quota for details.
Save your files to SharePoint Online team site when you want to share and collaborate on files regularly. You can also use team sites to grant permissions on site basis to limit the scope to only those people who are working on a project.
Save your files to OneDrive for Business when you don't plan to share them, or if you plan to share them with just a few people for a limited time.
Read full article.
Read Customize your Office 365 team site for file storage and sharing for detailed instructions on how to set up team sites.
No, you can still keep your files on your local drives and servers, if you choose to.