I am in the military and was deployed during 2016, can I claim storage fees?

Yes, storage fees during military deployment are tax deductible as moving expenses. 

TurboTax will ask you why you moved... the program doesn't distinguish between PCS orders and deployment - indicate that you are a member of the armed forces and moved due to PCS anyway - to enter storage expenses (see pic).

Turbo Tax data entry:

1. Type moving expenses in the search box (see pic) and hit enter - you must be logged in and IN your tax return
2. Select Jump to moving expenses blue link

OR:

1. Federal Taxes
2. Deductions & Credits
3. Other Deductions & Credits
4. Moving Expenses - follow the interview questions