Yeah, spiders are scary. I think they originally came from Mars or something. I have screamed like a little bitch on more than one occasion because of a spider no larger than my thumbnail was on me. We have some pretty good sized spiders around my place, but as long as they stay off of me and out of my house, I'll let them live.
Actually, I'm facinated by them at the same time. Their webs especially are incredible. Here is a good shot of one that was between the eves of my roof and the staircase on the outside of my house. Yes, I had to walk under this beast to get inside. I hate wrecking the webs too, because I know how good spiders are, despite my fear of them. For reference, the size of it's abdomen was about the size of a nickel.
As for the shed, definately attempt to spider proof it. If I was doing a bench press and I saw a spider crawling on my chest, I'd flip out and probably drop the bar and hurt myself WAY worse than a spider ever could.