The "magic" phrase is Network Attached Storage (NAS). That link will give you a description (way more detailed than needed). A web search for that phrase will find lots of specific units. I have not yet used one, so cannot give any recommendation.
Another option, which could be used with or instead of a NAS is a cloud service. Depending on your needs there are free ones provided by Apple, Google, Microsoft, etc. I use a for-pay one called SugarSync that syncs the files on my Windows computers, and also serves as a "NAS in the cloud." Downside of that is it may also sync for Android, but not for iOS and Linux distributions, and sometimes fails to sync properly on Windows.
Any storage option requires backup(s) for data you do not want to lose.
Another option, which could be used with or instead of a NAS is a cloud service. Depending on your needs there are free ones provided by Apple, Google, Microsoft, etc. I use a for-pay one called SugarSync that syncs the files on my Windows computers, and also serves as a "NAS in the cloud." Downside of that is it may also sync for Android, but not for iOS and Linux distributions, and sometimes fails to sync properly on Windows.
Any storage option requires backup(s) for data you do not want to lose.