If you get a quote at Safestore, you will get a £10 packaging and boxes voucher in your email. I've used it before and you are under no obligation to buy storage. If you feel a little guilty however, pay a £10 refundable storage reservation deposit and pop back and let them know you don't need storage to get your refund!
I've stored with them in the past and they are pretty good. They also have a price promise, so make sure you check out their competitors first.
If you do actually need storage, you can get 3x the value with Tesco Clubcard Exchange.
If you're a new customer, you can also get 50% off for the first 8 weeks.