Put in a bunch of IKEA dressers, then top with a countertop of your choosing. Here's one version that might give you an idea. Would be pretty inexpensive, but might still be worth getting a carpenter to trim and finish it.

http://www.ikeahackers.net/2012/03/koppang-dressers-plus-shelves-for.html

You could put three Ikea MALM 3-drawer chests side by side and top them with a countertop made of painted MDF or plywood.

I'm in California and I would suggest going to either Home Depot, OSH, or Lowes and buy the pieces yourself to put together. It's a bit more costly than Ikea, but from my experience, Ikea doesn't last as well as the product they have at Home Depot , Lowes or OSH.
You can also try and go on Craig's list and find the older style drawers that have about 9 drawers. That maybe be the most cost effective if you are willing to put in some elbow grease. Good Luck.

Ikea kitchen cabinets have drawers, too. Depending on the width of the cabinet frame, you can do 3-4 drawers. Going with simple white fronts and simple, sleek handles would give you a very close look. I have a friend who did Ikea base cabinets in her walk in closet and she's been pleased with the functionality.

As already mentioned, IKEA offers several options for recreating this look, at various price points and configurations. Malm dressers, Akurum kitchen units, or what I used, the Besta series cabinets.

others have suggested using different materials to top dressers/cabinets. i made an audio/tv cabinet using ikea countertop (numerar) and it's very nice; solid wood actually.

I second the Ikea besta cabinets, from the tv section. I have the taller white ones and used the same style of handles and the effect is very similar.

I second the Ikea hack idea. The Nyvoll chests have a similar look:

http://www.ikea.com/us/en/search/?query=NYVOLL

These look exactly like ikea base cabinets (as others have mentioned) with a countertop added. Instead of using legs that are visible, it looks like they used the plinth (kickplate see in many kitchens) which gives it a more built in look.

http://www.ikea.com/ca/en/catalog/products/S99869647/

If you look close I think they use these handles

http://www.ikea.com/ca/en/catalog/products/80153659/

but you could use these ones for a thinner profile (I used these in my kitchen)

http://www.ikea.com/ca/en/catalog/products/10226211/

And for the white square countertop you could use:

http://www.ikea.com/ca/en/catalog/products/70222724/

Or something with a wood edge: http://www.ikea.com/ca/en/catalog/products/00107671/

If you decide to use IKEA drawers, the company sells filler pieces to match the cabinetry. You use those to make the units look as if they are built-in from wall to wall. You won't have to jerry-rig anything. To make the kitchen units look more appropriate for a bedroom, I would avoid a kitchen-style top and use something that looks more like furniture. It's possible IKEA would have a matching side panel (for exposed end units) that would exactly match the cabinet material that you could cut into a top. In my third kitchen, I bought panels to make a refrigerator appear built-in that were about 8' x 3' and in the same finish as my cabinets.

I would like to dispute the notion that kitchen cabinets from HD, Lowe's or the like are better quality than what IKEA refers to as fitted kitchens. I have remodeled three houses using IKEA kitchen cabinets I put together myself and then had a contractor install. In each case, the contractor expressed amazement at the quality of the cabinets given what I paid.

I installed my first kitchen in the late 1980s, purchased from one of the first IKEA stores to open in the U.S. When I sold that house 13 years later, the Realtor featured my kitchen in the ads. The white cabinets were still in perfect condition. Not a wonky door, loose drawer or broken handle in the lot.

In its early days, IKEA was criticized for poor environmental quality of its cabinetry, but they addressed all those issues decades ago. Today they adhere to current European green-building standards, which are stricter than those in the U.S. And, no, neither I nor anyone in my family is employed by IKEA.

On an episode of Sarah's House on HGTV that I saw on Saturday, she built something similar using kitchen cabinets from a big box store. She wanted the piece to look like furniture, so it didn't extend from wall to wall. I do not believe she put another top on them.

Ditto Stream13. When I first saw the pic I thought they WERE Ikea base cabs.

Check this link referring to the "fauxdenza" on AT last year.

http://www.apartmenttherapy.com/smart-storage-solutions-for-sm-155053

Thanks for the feedback everyone! We were definitely thinking of doing something with IKEA and the ideas here are helpful. I realized after posting this, too, that the perfect IKEA solution lies in the GODMORGON bathroom cabinets. They are a little pricier but I think they would give precisely the look I'm after. Here's a great example:

→ Office storage with GODMORGON drawers at IKEA Hackers

Thanks again!

^ Very cool! You can also buy the kitchen cabinet top in white to achieve the exact look of the inspiration image.

this comes to mind:

Fauxdenza

I also was going to suggest IKEA, but also judging by the picture above (if your situation is similar to this) I would also place closet space on the sides of the window to maximize the storage space (so you have a built in look on that entire wall where the window is). Build the lower part for hanging shirts (half size items), and more shelves all the way to the ceiling for extra storage. I am sure ikea has a solution for this as well....

I could build them :)