Most of the features of macOS Server are being deprecated, including mail services.
Mail and other services are presently available on Server, though various of these will be removed with a future update.
Shared storage can use either a Mac or a network-connected storage device; what's called Network Attached Storage. macOS can serve its local storage to the network, and can for instance serve a locally-collected Thunderbolt storage array. Synology is one vendor of Network Attached Storage, and there are many others. You'll want gigabit networking, if you follow this path.
With the multiple displays connected to one system, you're envisioning a Linux system or some other classic timesharing system, and with multiple displays all being driven from one system. No Mac has enough expansion slots to drive all that; not with any sort of performance and any sort of resolution. Closest analog to classic timesharing these days is probably a Linux or Windows Server box or maybe a BSD server box with multiple thin-client displays. Those configurations very likely won't involve macOS, though.
One of the more common approaches for these sorts of situations in recent years is Microsoft Office365, the Google analog, or other similar software-as-a-service provider. That moves all your hosting and mail and most of the IT effort involved to the provider's servers and staff, which may or may not be desirable in your particular situation. This won't work if the data has to stay local for any of various potential reasons.
Beyond the soon-deprecated macOS with Server.app, local self-hosting options include packages such as ownCloud, and various others. Synology servers can also have add-on network services extensions, as well. I'd suspect some other NAS vendors have similar offerings, though haven't investigated those.
If you're considering running a local server, expect to have to allocate time and on-going effort to maintain and monitor and update and secure the server and the local network. NAS servers that are exposed to the 'net can and have been targets of attacks, for instance. Plan and budget for on-site and off-site backups, as well. Likely also RAID for critical data, though recall that RAID is not a backup strategy.
Have a chat with the business folks at the local Apple Store, and they should be able to provide you with some of the options and alternatives available when using macOS. Or scrounge up a consultant that deals in these cases.