sounds like you want tech called
Network Attached Storage aka NAS
The device itself is sometimes called a NAS engine
All the big manuf make engines (dell, emc, etc) also smaller guys like linksys, d-link, smc
an engine is nothing more than a simple stripped down computer that runs (usually) an open-source OS like linux or *bsd and acts as a network server... so you use almost any old computer (486 or better) into a NAS engine, which would also be a 'server'
if space savings is your interest, then look into Crusoe or Eden powered mini-itx systems... they're small enough to fit in a shoe box and draw only a few drops of energy.