Shared links in document

When you say "links" are you referring to graphic links and InDesign documents? 

You can use the same OneDrive, GoogleDrive, DropBox, Box, etc. account. (I know GoogleDrive allows multiple accounts on a single desktop, so users can still access their own account at the same time. OneDrive allows a personal and enterprise account at once.)

The desktop software for those services will sync from one's computer, then to the cloud, and then to other's computers. You just need to make sure you allow enough time to upload the document(s) to the cloud and back down to the computers again. This can take just a couple of minutes or longer depending on your connection speed.

@Spirited_power8696

You all can't work on the SAME INDD document at the SAME time - you need to split it into smaller parts - then use Book option. 

Or if it's just about links - then you need some cloud service or a server - as per @Eugene Tyson advice.

Immediately comes to mind is Creative Cloud Libraries, you store links in a Shared Creative Cloud folder, when someone updates a link it should sync automatically for everyone. 

Dropbox, Google Drive or OneDrive

You can setup shared folders - I think you need to  use relative paths. You can do this by having a Links folder alongside your InDesign file and InDesign should reference the Links folder. So your folder structure on the Cloud drives should be something like
<Folder>Magazine_Project/
InDesign_File.indd
<folder>Links
..............Image1.jpg 

..............Image2.png

Careful that only 1 person updates the links as could cause conflicts or have a filenaming convention - maybe if someone is editing links they could communicate that and come up with a system between the 4 of you.

Take it you don't have access to a server? Probably not...

A shared network drive would be handiest.