I included what look like consumer VHS tapes in the image but that's not necessarily what The Doctor is referring to. (Although some Pertwee Doctor Who episodes were recovered from what must have been consumer Betamax and VHS tapes, and/or U-Matic.) The 60s Doctor Who episodes were recorded to reel-to-reel videotape, which were wiped, and also rerecorded to film prints for overseas sale, some of which survive.
All of the first four Star Trek shows were shot on 35mm film. TOS, The Next Generation, Deep Space Nine and Voyager. Recreating DS9 and Voyager in HD or 4K would require a budget for editing and new special effects, as was done with TNG.