Records management is an ever-changing field, and as such, there are several options you can choose from when you decide to finally take care of all those boxes that have been stacking up in your office.
In your own city, you can probably find a place specifically devoted to records storage. At its most basic, records storage is just like the kind of storage you'd use to store your old futon or vinyl records.
Your records will be stored in standard "banker's boxes," generally provided by you, since you presumably already have the material ready to be stored. Then they are placed in warehouses or your own individual unit (much like a standard storage facility). Most facilities offer some sort of barcoding or other means by which to identify your records. Generally the identification and retrieval of your items is done electronically, as the way most everything is done these days. The technology allows these companies to maintain accurate records of not only what you're storing, but who accessed it and when.
Places that cater specifically to records storage generally advertise that they have extra security to keep your business documents safe. Some offer password-protected online access, and others allow you to make a list of the only people allowed to access the data. They screen employees thoroughly before hiring, they monitor the premises, and they have up-to-date smoke and fire protection in place. Basically, if you choose your offsite storage facility carefully, you'll be able to rest easy, knowing your documents are secure.
And if you're storing digital media, facilities will keep that material in a climate-controlled environment with backup power systems and other security measures that will ensure that your records are maintained at all times.