Can I replace a 3x8 1999 Sora crankset with a Claris FC-2403? I expect I would need a compatible bottom bracket too, like the one that this Claris crankset is frequently bought together with, here. The front derailleur is already a Claris, a few months old.

This is on a Jamis Aurora 2000 that I have been riding since 2003. Between my aging and the bigger hills here in San Francisco, where the bike and I moved two years ago, the gearing on the Sora, at 52/42/30, feels a bit too tall anyway. The Claris at 50/39/30 might suit better.

Would this be worth doing? The bike fits me well, I'm fond of it. The stock wheels are aging and the cantilever brakes are squealing, but it's still a fun ride. I'd have this done by the bike shop, I'm no mechanic myself.
Well, one consideration is that this particular Claris crankset (2403) mates to an Octalink bottom bracket.  The particular model of Octalink BB required is popular and as a result costs about twice what many others cost. 

Additionally, if you don't change your cassette (&amp; it's always a good idea to pair it with a new chain), you're really not getting any easier gearing on the low end as tooth count on the small ring remains 30.   The cost of improving your ride is becoming significant and the value of the new crankset alone is near-equivalent to the Jamis' value (looking at numbers from Bicycle Blue Book).  

A new road bike with a Claris groupset can be bought for around $500. They have compact and sub-compact options for cranksets and paired with a cassette with a tooth count  range of (for instance) 11-32, there's plenty of low gear for the hills.  Good bike shops will have staff that will help you with a good fit.  Take lots of test rides too. That's the best way to discover the bike for you.