You have two different ways in which you can close out your account. the first way is not to use it for 4 months and Yahoo! will deactivate it due to inactivity.
http://help.yahoo.com/l/us/yahoo/mail/or…
The second way you need to read the "Account Deletion" page thoroughly. When you delete an account you are deleting your complete ID. So that means you are losing Yahoo! ID, Mail address, and Profile names. We will also delete your account data and settings across the Yahoo! network. This includes the contents of your HotJobs, Yahoo! Mail, Yahoo! Address Book, Yahoo! Small Business, Yahoo! GeoCities, Yahoo! Briefcase, My Yahoo! and more. Account deletion will also terminate any paid services that you own with this yahoo id. You may not be able to recover payments remitted for these services or the data associated with these accounts.
If you would still like to terminate your account you will need to enter your password confirm your identity. You will then be asked to enter the code. Click the "Yes Terminate This Account".
http://help.yahoo.com/l/us/yahoo/edit/ch…
FYI--Your Yahoo Answers will not get deleted. This acccount is not in your PC. It is kept in a totally different data system. If you would like to keep it private you can do that.
You need to click your avatar or My Profile then you click the Edit Info. There will be another page that comes you and you scroll down to Privacy and there you can select the following:
Yahoo! Answers Network
Allow other Answers users to be my fan.
On my public profile, hide my Q&A from people not in my Answers Network
On my public profile, hide my Answers Network from people who are not in it
You can then click preveiw and then ok.