How much disk space does Yahoo Mail, Gmail need to provide such large mailbox?

Google uses a very large system. Non-current figures indicate "15,000 servers ranging from 533 MHz Intel Celeron to dual 1.4 GHz Intel Pentium III (as of 2003)
One or more 80GB hard disk per server. (2003)
2–4 GiB memory per machine (2004) ". And that's just for the search, so you can figure Google may have something like 1,500,000 gigs of space total. Google has a custom-built OS, see the link below for details.

Gmail Disk Space