Obviously, you will need disk space for that data.
But there are compression techniques, for instance columnstore, and there is a special COLUMNSTORE_ARCHIVE which gives even better compression.
It sounds to me like a good thing that you have separate databases for this archive data, because then you can easily move those databases around without affecting the main database.
Whether you rant to run the archiving daily, weekly or with some other frequence, depends both on the archiving and how the archiving affects the main database, and maybe more the latter.