maybe bitcoin could have a sidechain that stored the forum. and people could get rewarded with bitcoin when they stored significant parts of it. like mining the forum.

instead of proof of work it would be proof of storing the forum.
A way how this could be achieved:
- Sidechain has a native token and a Bitcoin bridge (see
this thread for some existing mechanisms), let's call the pegged token "bridgeBTC". Mining could be managed via proof-of-work/merged mining, or alternatively a PoS system.
- Forum storage takes place on-chain (like on Steem) and those who store new parts of the forum (which weren't stored before) get a status similar to "masternodes".* Transaction fees however will be probably relatively low.
- The standard client proposes, for each bridgeBTC transaction, to pay a small part to those storing the forum. This fee is however voluntary and can be canceled.
This is still a quite "fun" solution I don't think will ever materialize. But maybe somebody can even make profit with a semi-centralized "BitcoinTalkCoin".

But how big can torrents be allowed to be?
You could store all subforums separately, then it will be much smaller already. And you could create yearly/monthly snapshots with only the new topics. In general I'm quite fond of the torrent/IPFS idea. It will also show which subforum is really the most popular, and which ones would see almost no interest in storing or even downloading it.
* To know which sections are legitimately "newly stored" could be a quite difficult challenge. It maybe is impossible - at least without the forum operators themselves acting like a sort of "oracle" confirming the new threads are legit. Or there could be PoS voting (a kind of DAO) which new parts are "accepted".