Grad school in science that takes non-scientific applicants?
Speaking for the US and as someone with undergrad degrees in the humanities who is now doing a PhD in the sciences, it is hard, but not impossible to make the transition.

First of all, your best bet is Masters programs. Unless you're somehow exceptional (or maybe just very well connected), you'd be hard pressed to find a PhD program to take you when you're completely unproven in an area. However, to start out, there are transitional programs out there (Boston University's LEAP program for engineering is one) to help students move into a science field. Ask around and search online; I'm sure there are at least a few others.

You can also take non-degree courses at local universities to build up coursework in your desired field. For those, most schools don't much care about your background as long as you're paying them. Once you've built up enough background/understanding in your field, you may have some luck applying to Masters programs. I've seen other questions on this site where folks have posted schools that accept students to Masters programs in the sciences despite non-science backgrounds. I'll edit this answer if I can find some examples.

Additionally, once you've built some classroom credentials, you may have some luck trying to work with a professor in his/her lab. Even if you're not a matriculated student, I know of at least 2 faculty who have taken on lab assistants from outside the university. It's not common, but it is possible.

In summary, it's possible, but it will take time, probably some money, and some dedication to make the change. You'd probably have trouble getting in anywhere with zero background, but with some patience you can build that background and get your foot in the door. Once you have that, it just depends on how far you want to go with it.