I've always disliked the 6 year and throw rule. There are documents that you won't need to keep so long and other documents that you'd be wise to keep much longer. HMRC rules are only part of the issue.
I tell clients to do "weeding" and to keep "permanent" files.
In the permanent files, basically never to be destroyed, are the big stuff like loan agreements, asset purchase/sale invoices, important customer/supplier contracts, solicitor completion statements, leases, etc. This is the same whether you're in business or not. You just never know when you'll need a document to prove something, and that may be years after the six year HMRC deadline passes. Very dangerous, in my opinion, to chuck everything just because HMRC won't need it.
As for weeding, this is a lot more fluid. There'll be some documents that aren't really "accounting/tax records" that you can ditch far sooner, such as delivery notes, orders, statements, remittance advices, etc which are commonly filed with invoices - thus making your invoice files much thinner at a stroke. Then there's the stuff which really is "accounting/tax records" but which is practically irrelevant and unlikely to ever be needed, such as smaller invoices, petty cash slips, etc - OK in theory should be kept, but if the figures are small, very doubtful it'll ever be looked at again. If you had, say, four files of documents, I'm sure you could weed it down to 3 by ditching the small/irrelevant stuff fairly quickly and easily. Do that for six years and you've reduced your file storage from 24 down to 18 files. Of course, don't do that too soon - wait at least a couple of years for the year accounts/tax to be done and another year for the standard tax return enquiry windown (typically another year). But, please keep the important stuff, such as bank statements, all invoices that aren't trivial, etc.
Payroll and employee records need to be kept longer than 6 years anyway - don't ditch those after six years. Also, some professionals will need to keep their files for longer than six years due to liability risks, i.e. solicitors/accountants - ok, they're not accounting records, but still a risk that they'll be destroyed after six years if you have an automatic 6 year and ditch rule which means you'll have no defence if you're sued!
Oh, and don't even think of ditching any of your "accounting records" themselves, such as accounting books, ledgers, computer reports, data backup discs etc - definitely six years minimum for those, and preferably to be in the "permanent" box.
But to the OP, no I can't imagine that lack of space would ever be accepted by HMRC as a reason for not keeping the documentation.