How can I obtain a complete copy of all 635 issues of The Spectator?

The Spectator has actually made it extremely easy for you to find archived articles from older issues of the magazine. In fact, The Spectator has a digitized archive, fully available online, in which you can find articles from copies of the magazine published between 1828 and 2008, 180 years of magazines. This archive can be found here.
If you want to search by issue, rather than by topic, you can find the articles sorted by century and decade in the issue archive, which can be found here.
For issues of The Spectator from 2002 to the present day, you do not need to visit the digitized archive, which contains scanned copies of articles from physical magazines. Instead, you can simply visit the drop-down menu on the far right of the main magazine site or click here.
Eventually, you may find you hit a limit on the number of articles you are allowed to view, in which case you may need to subscribe using one of these plans.
It is not possible to acquire physical copies of every issue of the magazine ever printed, but the archiving that has been performed by the magazine is extremely extensive, and if there is an article you are looking for that has ever been published in The Spectator, it is possible to find it using the links above.