Synology NAS devices offer a few of features that you mention: email [0], calendar [1], document sharing [2], chat [3], office apps [4]

Shift/inventory management would need to be added via third-party software (or implemented in spreadsheets).

I've heard of companies setting these up for small businesses. [5]

[0] https://www.synology.com/en-global/dsm/feature/mailplus

[1] https://www.synology.com/en-global/dsm/feature/calendar

[2] https://www.synology.com/en-global/dsm/feature/drive

[3] https://www.synology.com/en-global/dsm/feature/chat

[4] https://www.synology.com/en-global/dsm/feature/office

[5] https://www.synology.com/en-global/wheretobuy/United%20Kingd...