I moved to South London from Ipswich and needed a storage solution for the (many) things that I didn't have time to root through! I'm currently using Safestore in Mitcham, which is about £80 a month for the standard 25ft unit. They say this is the size of a small shed, but with some good Tetris-ing I've fit almost a 2-bed flat in (sans big furniture).
They will do some sort of opening offer (50% for the first 8 weeks or the like) and match prices as well. 24-hour access was important for me as I work during the day, so it's been great for me!