The company I worked with spent a non-trivial amount of money storing historical tweets. I'd even go so far to say that was the majority of the IAAS costs - even more than the compute required to process them in real time.
First, how many accounts are we talking about here? Between POTUS, VP, their spokespeople, cabinet members and official agency accounts, I'm going to assume the executive branch has about 50 accounts that ought to be archived. For congress, I think it's reasonable to archive each member's account and their spokesperson: 200 accounts for senate, 870 accounts for congress. In the interest of Fermi estimation (and because I'm not sure if every one of these people has an account) let's call it 1000 accounts.
I'm going to go with a mean of 10 tweets per day (again in the name of Fermi estimation).
With 280 characters + metadata, I'm going to round up to 1KB per tweet.
1000 accounts * 10 tweets/day * 1KB/tweet = ~10MB/day = ~3.65GB/year = ~40GB for the current lifetime of Twitter
If you're drinking from the firehose to archive tweets for a huge userbase (and feed them into models or perform semantic analysis) I could see this getting expensive and costly. If you're just try to archive tweets from a certain group of users and keep them on a disk (with a tarball or zip file released quarterly) it feels a bit more doable.
That said, this little thought exercise has gotten me thinking a lot more about what I expect of the LoC and the National Archives. I'd be happy with a "cold storage" record of the tweets being preserved for posterity, but they may see their mission as making the tweets into a tagged searchable collection. I also think there are arguments to be had about how many people's account really need to be archived (perhaps states could handle archiving their own reps).
In the end I guess I'm cool with the LoC scaling back collection as long as they're transparent about how they're doing it.
You can tweet photos, videos and GIFS. My average gif is 500kb.
If these were excluded I suspect it would make the archive much easier to manage.
The article states that it was archiving _all_ of public twitter. It is moving to the selectivity that you have assumed was status quo.