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I'm sure you've already found these, but by some margin the closest to you seems to be this:
http://leevalleyestates.co.uk/directory/business-listing/storesafe/
(storesafe.com redirects there.)
I guess it must be containers plonked on this large empty site until it gets redeveloped:
There's also Shurgard in Highams Park, which would tally with Dov's recommendation:
https://www.shurgard.co.uk/self-storage-uk/london/chingford-highams-park
(Oops, missed your reply to Dov.)
There's a self-storage place in Leaside Road, both very close and not so close (because you can't exactly go as the crow flies) to your new place. I can't recommend it, but a friend had some of his stuff there for a while.