I've had a cluttered Inbox for as long as I can remember. I'll occasionally forget to follow up on something or not notice an email at all amidst the mess. If I had known it could be so easy to get everything under control, I would have done this ages ago, but I never took the time to look into some basic features of Gmail until recently! I had thought that the only way to do it was to, well, delete everything.
Using Gmail it is incredibly easy to clear out your inbox without losing your previous emails. Search for label:inbox, select everything on the page, then click "Select all conversations that match this search" and click Archive. This will move all of your email out of your Inbox and into the All Mail tab, where they will forever remain. It's that easy!
I would recommend, while you're there, to spend a bit of time clearing out the obviously unneeded messages - for example, I had hundreds of Trello notifications and a lot of exception notification from web apps that I work on, and I just searched and did the "Select all conversations that match this search" to delete them.
How I plan to keep it empty
E-mail is pretty good for communication, but awful as a task manager and general repository of knowledge. So now when I receive an email, I now try to decide immediately where a better place for the information would be (a todo list, my calendar, Workflowy notes) and put it there. If it's something important and I'm not sure where to put it, or if I need to reply to it, then I star and archive it, so I can easily find it but it's outside of my Inbox.
I also use unroll.me to roll up unimportant but potentially interesting emails into one daily message, so I have less to process throughout the day and far less to delete.