My understanding is that it's a giant BBS, just a large archive of newsgroups.
Your understanding is wrong. Usenet is not a single bulletin-board system. It is a network of intercommunicating systems. It's also — much to some people's dismay — not an archive.
Is it mostly just old-timers?
This is one of the parts of the question mentioned above. The participation varies, both from node to node and from newsgroup to newsgroup. It's impossible to say, not least because it's impossible to even identify some posters, but more importantly because no-one reads the whole of Usenet. It's — still — too large to be read in its entirety. Anyone who tells you what "most posters" are is telling you about xyr own small corner of Usenet, and is probably wrong.
Why would I need to pay for access to it? Aren't there sites that aggregate it for free?
One has always paid for Usenet and other network access in my experience. It is simply that one wasn't always aware up front that one was paying. My ISP, several years ago, used to provide a Usenet node for its customers as part of the package, for example. The cost of Usenet access was incorporated into the general cost of Internet access. Nowadays, of course, it, like many other ISPs, has long since discontinued its own Usenet node.
Yes, one can run one's own Usenet node. I do. But anyone who has done more than play at running a Usenet node will tell you that there are costs involved.
Why do I keep seeing ads for cheap terabytes (!?) of storage space on usenet?
Because to some people Usenet is a glorified file transmission and sharing medium. You'll hear people babble on about "retention" of binaries newsgroups, and boast that their retention times are better than those of the next fellow. That's all about using Usenet for file sharing.
(We Usenet nodes who participate in non-binaries newsgroups don't bother with that nonsense. Many nodes, including mine, effectively turned off article expiry for non-binaries newsgroups half a decade ago. The size of non-binaries traffic compared to binaries traffic is so vanishingly small that it's basically lost in the noise of a full Usenet feed, and non-binaries traffic isn't what it was in the 1990s in many hierarchies in any case. Power Usenet, for example, is currently advertising that it hasn't expired non-binaries postings for over eight years.)
People post chopped up, encoded, files using multiple incompatible encoding and partitioning schemes, in newsgroups where messages can arrive out of order and sometimes not at all, and other people spend their time trying to piece back together the results. There's a whole cottage industry of softwares to "help" people both post files to, and read files from, Usenet binaries newsgroups.
Yes, the world has invented better ways of distributing files since the 1980s. It has also invented more user-friendly and non-English-friendly discussion forum mechanisms — that, for example, allow people to converse using em-dashes; hyperlinks that don't break at the drop of a hat; quotation markup (such as the <blockquote>s in the hypertext of this very answer) that is unambiguous, works reliably, and is not hypersensitive to line wrapping activity; and (gasp!) real smileys. ☺ Some people still use Usenet, though.
.planfile, didn't you?), telnet ... – dmckee Oct 26 '11 at 3:06