"When you mean 'finding somewhere for your auction software to run' do you mean a hosting service?"
Yes, it would need to be hosted somewhere that supports applications and database storage as well as allowing enough traffic to the site to support the users.
"And how, do you think, I could find a developer who's competent but also used to dealing with very small businesses?"
That I'm not sure how to answer, I've always been the developer rather than someone trying to find me :)
Your first step might be to see what you can manage on your own. If you lack IT skills then a hosted solution might be best as that will take care of a lot of your day to day support worries and will mean that you won't need a techy permanently on hand and can rely on the support from the hosting company to help you out. But yes that does mean if the hosting company collapses you're left in a world of pain....
I recently look at these guys:
www.rainworx.com/
Have a look at their online demo through the admin, buyer and seller modes.
Once you see what people like that can offer you out the box then you might get a better understanding of what you want to customise, and that in turn will let you find a developer and clearly state to them exactly what you're after.
"Another question that even I am astonished I don't know the answer to - what's a server?"
The actual computer that something is running on, so think a big beefy Windows machine that exists somewhere out on the internet.
I'd still think very hard about trying to go up against eBay, why will people be drawn to use your site and not eBay?