It's even worse - the paper used nowadays is not very robust - in older days stone or parchment (or more stable paper) - which could last centuries. Even the paper used in our days will only last 10-100 years - and thats not long. Though: not only our digital data, also the analog is going down. Our houses are not build as strong as the fortresses from older days - and who is nowadays throwing corpses and treasures into swamps?
At least plastic seems to be robust
And .. well - to the BBC: It should be known that "data archiving" is not finished with backuping it once - the backups have to be checked, copied and reformatted every 1-5 years ... (It's not done with ensuring that you can read the media - who could read '88s data format? Which is why some special (open-)formats (e.g. special pdf types) are used for long-term archives).
OTOH: no worries, most governments use some kind of long-term archives (e.g. in old salt mines, saving data on microfilm or (not joking) graved into copper plates).