To share your Gmail mailbox you need to enable Email Delegation in Gmail and to enable Gmail delegation follow the below steps.
Step1: First of all login to your Gmail account.
Step2: After login click on the Option link (The geared icon on top right corner) and then click on the Mail Settings. Now navigate to the Accounts and Import by clicking on the tab.
Step3: At the bottom you can see grant access to your account, there you can add the email id with which you want to share your mailbox. Thats all, now the other person will get an email with a link to accept or reject the request. Once the person accepts the request they can send, delete, and receive emails on your behalf. Word of Caution: Before sharing your mailbox be careful to share it with the person whom you think the faithful one. While entering the email address in email delegation, be careful and double check the email address to avoid any mistakes. However you can delete/disable the delegation at any point of time.