Your picture seems to show 2x4s, a doubled sill plate, a gap, then a header. If that header spans from an exterior wall to an exterior wall, the 2x4s and doubled sill plate aren't doing any work other than holding up drywall.

Wow. I’m just gonna be straight. I’m very offended. I have no plans to do this myself. This started as just a question about COST. I get that it might be difficult to assess over the internet. I NEVER said I was doing the work myself. I WILL BE hiring a contractor. In fact I mentioned that one is coming on Monday. I was just trying to COST things out. I’m not an idiot. I don’t base decisions on HGTV. WTF? Just to help commenters in the future ... maybe you shouldn’t make assumptions about the OP’s motives, plans, capabilities, thought processes... I got some good info here but I really wasn’t expecting to be insulted in the process. So I’d appreciate it if everyone just dropped it. Pretend you never even read such a stupid question as “how much does this cost” so you can all sleep at night. Oh, and for those of you that think I’m an idiot for wanting to open the wall, please don’t waste my time. I asked about cost. Heck, I just expected someone to say that either there’s a huge cost difference, there’s no cost difference or it’s not possible to know without seeing it in person. I did get that last response and some great info from a few folks. But then I was called an idiot, that I lack a multitude of capabilities, that I base decisions on HGTV, that I have horrible tastes, etc. Wow. Maybe I just don’t know the etiquette. Is it okay to ask questions on Houzz that I don’t know the answer to? Is it okay to not be a professional contractor? Is it okay to not know what I don’t know? Maybe I’ll update this thread after I’ve destroyed my house and I’m living in a shed. Because I’m just that stupid, right?

A nook for a stove won't work very well. You need a minimum amount of clearance on both sides for the stove to be functional. And in fact, many manufacturers have minimum clearance requirements to meet fire code. This is usually on the order of about 12" on each side, although in practice that's really tight and awkward. I'd recommend at least 18" and preferably 24". Your drawing makes it look as if you won't have sufficient space to do that. So, you probably wouldn't be able to get the work permitted -- and even if you did it without a permit (which I won't recommend), you'll suffer from a poorly functional layout. The overall layout of the kitchen looks pretty awkward to begin with. I love cooking and like a very functional kitchen, so I probably wouldn't bid on the property because of that. Your situation might be different, depending on how important a good kitchen is to you. Having said all that, cost depends on so many different factors. It's really hard to say. It could be as easy as installing a new header, repairing a little bit of sheetrock, and patching the floor. Depending on what materials you choose and how expensive your local labor market is, I could see an absolute low-end price of around $2000 if hiring a handy man (not a great option when doing structural work!). But in a HCOL part of the country, with higher-end materials, and maybe a complication or two, it wouldn't be unusual to cost $20,000. And if you have to involve a structural engineer, an architect, and a few iterations with the planning department, you could even double that cost. If the city makes you bring things up to code in the process, things could balloon even more. The good news is that while all kitchen remodels are expensive, once you get to the upper end of the price range, there only is a small incremental difference between moving a wall and doing a full remodel. So, the good news is that at least you'll get a lot more for your money; the bad news is that until you actually get quotes and maybe have somebody open up walls, it's really hard to give you any firm numbers.

My guess would be $20K-$50K. A lot depends ....