Outlook can rely on local data files called PSTs for archiving emails. Sometimes these PSTs can get corrupted.
In the unfortunate instance that they stop working, you can use a tool called “SCANPST.exe” that ships with Outlook to try and repair it.

1. Navigate to C:\Program Files\Microsoft Office\Office15
2. Locate the SCANPST.EXE icon and double-click it.
   
3. Locate the broken PST and click on “Start”.
   
4. The tool will take a while to scan the broken PST.
   
5. Then it will tell you what’s wrong with it and ask you if you wish to repair it.
   Click the “Repair” button.
   
6. The tool will work away (you will probably get a “non-responsive” status until it’s finished, but ignore it and just let it do its thing).
7. Finally you should be presented with the “Repair complete” dialogue box.
   
8. Open Outlook and see if your PST is now accessible.