I can heartily recommend Access on Willow Walk (Grange Road). Clean, dry and secure in a modern purpose built shed, and the staff are excellent. Self storage is not cheap, mind. 200 sq ft = £440 per month, thus making it £25 psf per annum. Cheaper to rent a flat in Deptford...
I can also Mapmaker's recommendation of Access in Willow Walk SE1 - I used them for 4 months last year. They were very flexible in terms of unit size (I needed bigger than I originally anticipated), cost & time - even reimbursing me for leaving early.
I think Juanita's point is that if you move out (having given the required period of notice) you'd get reimbursed for the unexpired part of the month/4-week period, as you'd already have paid the rental cost in advance.
That's certainly the way it worked at Safestore Deptford when I had a unit there.