The great thing about a closet is that you can simply close the door to hide the things that you don't keep super organized. The key to organizing a room without a closet is to make use of all the hidden nooks and crannies your room has to offer; or better yet, make your own. If you're using open shelves, it's worth it to make a small investment in purchasing attractive bins and baskets that match. That way, regardless of how unorganized the items in the baskets are, from the outside, the space will always look neat and tidy.
The area under your bed contains a ton of unused space that's perfect for storing bulkier items, like sweaters and linens. And since you can't really see under the bed, you can just buy the cheaper plastic containers. Opt for clear so you can easily see the contents, and it doesn't hurt to list them out line-by-line on a label, too. You can also stash craft supplies, rarely used dishes or pretty much anything that you don't use that often but would like to access quickly when you need it. Wooden chests placed at the end of the bed also provide attractive storage options -- and can be a family heirloom passed down through the generations.
Another solution that won't break the bank is to get stackable clear containers that fit in the corner, and then hide them behind a room screen that's easy to move aside. You can also line the top of your walls near the ceiling with hanging shelves that will hold inexpensive but colorful cardboard boxes that are great for storing smaller, lightweight items, like socks, lingerie and jewelry. Then, just keep a little fold up stepping stool on hand for easy access.