We are needing to store data somewhere that multiple computers can access. Is Sharepoint an option if we dont want to use ArcGIS online? How to set up if so?
How much data? And is this a static set of data or something that will be subject to changes?
Is this spatial data or something else?
Mostly spatial. But could be anything GIS related.
Static, non-edited data might be ok on SharePoint. Just make sure you have appropriate attribute as well as spatial indexes to improve drawing/query speed.
If you are looking at multi-user concurrent editing then SharePoint may not be the best bet as it is not much better than a standard network share from a database perspective. You really should consider a backend database like MS SQL, Oracle, PostgreSQL, ... that can adequately handle the multiple edits and present data in an unlocked fashion.
File geodatabases can be used for multiple edits on SharePoint (or other local resource) IF everyone stays in their own area. https://support.esri.com/en-us/knowledge-base/can-multiple-users-edit-a-file-geodatabase-at-the-same...
The best would ultimately be some form of database as @Thomas3 suggested. This will cost money in cloud resources, but for the amount of data and editing capabilities you would need, this is the way. (unless your college has some resource where you can deploy databases on their dime.)