My school has tasked me with turning them into a Cloud server

My school has tasked me with turning them into a Cloud server

My school wants to have a Cloud server. I semi-understand how Cloud
servers work, but I'm not sure how they'd be able to do this. My school is
small (about 300 people from Kindergarten to Grade 12) and I've been the
one to help with "tech support". No one really has a clue about computers
and I'm kinda experienced with them. They've asked me to "make them a
Cloud server". Please help.
My question is, how would I go about doing this? The school has private
servers. On all computers in the school, when you go to the "Computer"
folder there's a Drive called Common (Z:) and student login (C:). The
Common (Z:) folder is accessible by any student, and it's usually the
place to hand assignments in. student login (C:) has all documents for
that particular student and it's accessible from any computer but only for
that user. My school wants to be able to go to a website or open a program
from any computer not on the network, enter login credentials and be able
to view student login (C:) and Common (Z:) so students can hand in + work
on assignments without having to bring a flash drive or CD.
Any help would be appreciated, I don't know what I'm doing.