A box spring is a shock absorber for your mattress, but it's not really necessary. If you have an innerspring mattress, you're just double-sprung. If you have a memory foam mattress, the salesman probably told you not to use a box spring. A box spring also makes your bed higher. A mattress-and-box spring set usually comes with a metal frame on legs, which makes your bed higher still. If you feel like a princess on a pea and get nosebleeds in your sleep, consider some homemade low-altitude alternatives
Build a plywood or fiberboard platform to support your mattress. Nail, screw or bolt sideboards of any height you like together to create a box the size of your mattress, add as many crosspieces as you need to give your mattress good support and top it with a solid sheet of wood cut to fit. If you can get an old waterbed platform, it's ideal for the purpose.
If you have ten thumbs and no carpentry skills, get hold of some plastic milk crates, but be sure they are all the same height. Six should be enough to support a single mattress, more for a larger one. Top these with wide slats of plywood or fiberboard. A home improvement center will cut these for you. This alternative works well if you move a lot – you can pack stuff in the crates and the wood pieces are held in place by the weight of the mattress.
If you have a board, but no milk crates, use cinder blocks or even bricks to support your bed board and raise it just a little off the floor. You may want to put something plastic between the bricks and the bed board so that if the floor gets wet, the bricks don't soak it up and wick it up into the wood and eventually into your mattress.
As long as underbed storage is not a primary concern and you have a nice, flat floor, you can put your mattress directly on it, even if the floor is concrete. It can make bed-making a little inconvenient, but if you fall out of bed, you won't get hurt.