Before microfilm was common, back newspapers were stored in a "morgue." The newspaper itself would store back copies, usually filed by date, in boxes (laid flat, folded as if on a news stand). This was the case in the United States from the time papers started to be more than a broadsheet until microfilm became cost effective (around WWII, I can't tell you an exact date), and I believe the practice came here from England.
Larger libraries would also keep back copies of papers, but not for very long -- a month, two months, perhaps a year for a very large library (and then likely only for major papers like the Times). This was mostly due to space limitations, but also because there were so few request for older papers to make it not worth saving the older numbers. A newspaper, by contrast (barring fire or other disaster that destroyed the morgue or its contents) would keep every edition from opening day until they closed the doors, and at that point, the old papers would often go into storage with one of the principles of the firm.
Many papers (and libraries) had microfilm only for dates after they started filming the archives into the 2000s, having never gotten around to filming back into the older numbers.