Tool box help!
The bed on your 2013 F150 is 22.4" from the top of the bed rail to the bottom of your bed. The bed is 65.2" inside to inside of the bed rails which likely accommodates the overall width you are looking at. It looks like this will fit. Most toolbox manufacturers have a fitment guide on their site which will ensure that your toolbox will be compatible. Now that said if you are looking for a toolbox that you can access all of your gear without crawling into the back of your truck or needing a step ladder to reach into, we've got a product you might like as well. Toolboxes like you are describing work great, but sometimes not the most practical or useful. It all depends on how you use your truck.
Good luck!

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Ford specs: HERE
It should fit. Also I'd recommend getting a "better built" brand toolbox hold down kit so you don't have to drill holes in the bed and it'll be held down good. you can get it at most Rural King stores and some auto places


