TL;DR: Backup (of local data) at the time the show is set was mostly done to floppy disks, and even when using a network, all was at the mercy of the user to remember doing it (some companies tied backup into system boot, but then there are zillions of stories about users disabling it, 'cause waiting sucks).
Backup of centralized data was done independant and with whatever the mini or mainframe provided (usually tapes).
Being that the show takes place in the mid 1980s, what were their options for (potentially centralized) backups of production data?
Quite similar to today, well except large disk arrays that is. Tapes where the choice since the mid 70s (when they slowly came out of fashion as main storage). Since then they ofc improved quite a lot. Where originally 6250 bpi tapes where standard, the mid 70s brought the introduction of various cassette based tape systems. While there was quite some competition, the QIC standard did gather the largest market share in the non proprietary market (What a bunch of words to say that IBM didn't care and developed their own standards (LTO)).
During the early 80s, which is the seting for the show, four and nine track QIC drives where most common, used with 300, 450 and 600 ft tapes. Cassettes where named according to the tape length, so a DC300 held a 300 ft tape. Four track QIC drives stored about 20 MiB on a 300 ft tape, while the nine track variant doubled this to 40 MiB (plus better error correction).
At said timeframe these drives where mostly used in (UNIX) minicomputers and in top end workstations (Apollo for example delivered most machines with a QIC drive). For simple PCs, such drives where available but rather rare. Keep in mind that during the first half of the 80s, floppy-only machines were still common. It wasn't until 1983 that IBM introduced the XT with a standard option for a hard disk.
I guess that the part of the show where next to every machine has a HD is less historically correct, but necessary to make it belivable to today's viewers :)
Besides QIC, there where other standards which did get some audience during early PC times. For example DCC, working on compact cassete like drives, but using a digital format with special tapes. They could store similar amounts per cassette, but has slower access times.
So yes, single machines could have been equipped with tape drive (QIC or other), but it's rather unlikely that it got rolled out all across a company. In context of the show, some introduction of a company invented new device would make sense... much like the Iomega did with the Ditto drives in the 90s.
The standard medium for backup were floppy disks. From today's view this may sound impossible, when a single letter can have several megabytes, and a floppy just holds 360 KiB, but back then programs didn't use bloated markup. Even with a lot of formatting a word processor file stayed under twice the amount of visible characters. A rule of thumb for letters were about 2-3 KiB per page. So one floppy could well store a hundred pages without compression or alike. Quite workable for a weekly differential storage for a single PC.
Bottom line, while there have been solutions available even before the IBM PC got introduced, the real need for tape drives for single users didn't manifest before the mid 90s, when disk sizes and more importantly data formats increased exponentially in size.
There's some allusion to re-installing the backup drives; I assumed these were dumb, block-level clones taken when there was some extra slack in the network, not some good managed backup framework with fancy things like deduplication or application snapshots.
Well, I have to admit, I neither watched the show closely nor do I have an idea what kind of network / mini / mainframe setup is featured. But since the premise is about a PC company, it's safe to assume that there is a lot of workstation orientated application and data storage, so decentralized data (data stored on each workstation) needs to be backed up.
Considering a network with most PCs connected (rather unusual for the early 1980s) it's quite possible that there was some IBM mainframe with extensive tape storage used to backup workstations. Usually the backup would be made in two stages: first the single PCs would be backed up onto mainframe disks with separate areas for each user, and in a second step all these user backups would be exported to tape.
Most likely not.
The backups in step one would NOT be made up from simple block copies, but rather on file level and differential, meaning that only files changed since last backup will be copied. DOS supports this strategy since 1.0 via the Archive bit, inherited from CP/M 2.2.
taken when there was some extra slack in the network
Not in the early 80s and on PCs.
MS-DOS was a single user single task operating system. There was no room for a backgound process nor the necessary interfaces to do a backup without interfering with the user. DOS wasn't reentrant and the archive bit mechanics did not work well in a multi-tasking environment.
Backup was strictly a process done at will and never really in the background. Considering the meager amounts of data to back up, even over a slow network all transfer could be done during a day in a few minutes. So no need for additional complexity.
Of course one could assume they were using some kind of multitasking environment running every PC, but that's extremely far fetched, as Desq (later named DESQview) wasn't introduced until 1984 and at first was 'just' a switcher on top of DOS, thus not able to run an independent background process. The 386 changed this, but that again should be way beyond the time frame the show is set.
Similar for Novell's Netware (to consider the network side). Until Netware 286 V2.0 (?1986 or 87?) it was a strict add on to DOS, hooking some functions to redirect access to a specified 'drive' over the network. All activity was client initiated. Again, no background processes at all, thus none waiting for 'network slack' to do some background.
And yes, there are zillions of other posibilities and combinations, I tried to stay with the most common/generic.