I know that the popular opinion on this site is that undergrad doesn't matter at all, but I'll go ahead and disagree here.
It's certainly true that in the grand scheme of things undergrad doesn't matter nearly as much as your GPA, MCAT, extracurriculars et al. It is about what you do with the degree more than where you get it. I wouldn't dream of telling a high schooler to take on 160k of debt just to have a better name on the resume.
However, to some extent prestige begets prestige. This isn't pretty, and not nice to think about, but it's true. I went to a good college (top 20 USN I think). I am now at a top-10ish med school and I interviewed at a lot of the top places. Literally almost everyone at the interviews with me had gone to a college as good or better than mine. Some of them were state schools of course (the UC's for example) but for the most part we're talking ivies and thereabouts. This is because those were the students who had had fantastic advising by well-established and knowledgeable premed advisors, who'd had great research opportunities with some of the greats in their respective fields, often from schools with their own big medical center where they could shadow some bigwig who then the med schools would recognize. Not to mention the higher likelihood of them doing things like mission trips funded by the school, or winning a Rhodes or a Fulbright. It's true that biology and chemistry are the same everywhere, but at my school (which was by no means an ivy) after the intro classes I was able to take tiny seminars of 10 people or less with professors who'd done some great and interesting things and who were willing to let me work with them, get letters of rec from them, etc.
Again, you can absolutely succeed no matter where you go. But it's a bit of an exaggeration to say that it won't matter at all. My classmates in college were all extremely smart and dedicated and ambitious, and we studied together a lot. Our grades were rarely on a curve, and when they were there were often so many people who'd done extremely well that it rarely ever helped anyone. It was hard to get that A, and hard to hide in the crowd or let the dumb classmates bring down the curve- cause there really weren't too many people who weren't trying. Per my classmates who went to bigger, lesser known schools, their difficulty was in standing out, in finding great research opportunities, in receiving great advising. Obviously they ended up at the same med school I did, so they succeeded, but it was harder. Also, saying that a 3.8 from MIT is the same as a 3.8 from U Mass is just ridiculous. And no adcom would think that.
So all I'm saying is- don't stress out too much if you can't get into Harvard, you'll be fine. But it won't be nearly as easy to get to Hopkins from a place that's relatively unknown, so be forewarned. Of course you can make a great argument for why going to Hopkins itself is entirely unnecessary, but given that everyone here thinks they can join the faculty of Harvard plastics, I figured I'd play devil's advocate. If you have the opportunity to go to a really good school without totally breaking the bank, I don't think it's unreasonable to do so at all.