Depending on the actual loads involved, maybe you can't. But "can't" meaning "wouldn't" because of the costs involved.
At minimum, "B" has to be doubled (perhaps tripled) in strength, and posts added where it meets "A" and "C" which so-called "point loads" have to be transmitted to footers in the ground. If posts to the ground aren't feasible for you, then "A" and "C" have to be strengthened, including where and how they bear on the walls. The strengthening part, depending, again, on the loading, could mean replacement with heavier wooden members, or steel beams. The whole thing may need to be steel (A, B, and C)if the engineer determines that wood beams are not practicable due to the sizes required. IOW there may not be headroom for a 6x18 wooden member, but a 4x12 steel I-beam can be fitted just fine. None of us online would be qualified to judge any of these parameters, but we can help weigh alternatives offered by the engineer that you engage to look at the job and make their recommendations.
Casey

Your drawing and description lacks critical details: spans, joist sizes, joist spacing, joist doubling/tripling etc and any continuing supporting structures in the basement.

Before you remove anything other than wall finishes, have an engineer or city building inspector take a look at the situation.

Salami, with enough time and money your job can be done. I would imagine, your trying to do this for little money. It can't be done for little money. Its at least 4k per floor guestimate.

Thanks Casey for the advice. I am sure that with steel reinforcement, or possibly replacing A, B and C with LVL beams, this could be done. I will definitely consult an engineer first, but I thought gardenweb folks may have some good insight. Does my drawing indicate a low budget! that is funny. I am just trying to figure out the structural possibilities...

Unless the bearing walls are masonry you will need to provide a header or a post to distribute the new loads to the foundation.

Considering the concern of others about the cost of the project to be funny seems a bit snarky. The answer to your question is pretty obvious so it is natural for others to think you don't know what you are getting into.

I don't know! Thats why I am asking for advice! But I do know that some LVLs or posts or both don't cost $4000.

The bearing walls are in fact masonry, it is a c. 1870 rowhouse, and all the joists are about 3x9, pretty heavy duty. I think B alone can probably carry the load of a 10' by 10' floor system but I am looking for 2nd (and 3rd and 4th) opinions. Thanks for any advice...

The cost of cutting joists back to make room for a triple LVL girder and making the connections at the masonry walls could easily cost more than $4,000 unless you're going to do the work yourself.

Salami, unless your an engineer with a license you can't make these decisions, unless your calling from a location that doesn't have any building codes. Engineering plans alone will be at least $500 after you remove all the wallboard to expose the supporting structure. How much are you hoping this will set you back per floor?

salami, you are looking at a pretty simple system to engineer. You can even size your own lvl's using the manufacturer's span tables... http://www.parr.com/PDFs/LP%20LVL%201.9E.pdf that's the tables for LP. You certainly want to make sure that you are taking all loads into account, ie is "B" carrying a wall above it that carries the next floor? And you certainly want to make sure that the point loads at the end of "A" and "C" are properly supported to carry the new loads. LVL's aren't cheap, but you might have a few hundred bucks in them and a day of back breaking labor. You can certainly get away with out hiring an engineer. Often the lumber company that you purchase the LVL's from will have an engineer on staff to size them for you. Getting onsite advise from someone knowledgeable in this sort of thing (a building professional) would probably be a good idea too. good luck

Here is a link that might be useful: LVL span tables

In my experience, it is rare for a building department to allow a homeowner or even a builder to design a beam using a manufacturer's span tables (Maine & Vermont allow it) so an engineer's stamp is usually required. However, it is possible to get a stamp from the manufacturer through your local lumberyard if the structural conditions are not complicated.

If you use LVL's read the manufacturer's installation instruction very carefully especially if the beam is built up from multiple members and/or there is side loading. What appears to be simple and obvious often is not.

"The bearing walls are in fact masonry, it is a c. 1870 rowhouse, and all the joists are about 3x9, pretty heavy duty."

The issue with very old places with what are not 'non-standard' beams usually requires a PE stamp for approval.

You can run into all sorts of 'gotchas' trying to alter a structure this old.
The bearing points for any new framing have to meet the code in affect now, not in 1870.

Even a simple change can mushroom into a huge battle with the AHJ over spans and carrying capacity.

I have worked in even older buildings (pre-1776) and often the only thing possible is to replace with 'like material' with a PE stamp to leave the AHJ 'off the hook.'

Trying to find 8x6 oak beams is not for the faint of heart, or budget.

My parents had a similar situation when they converted their 1890s two-family into a single family. They removed the center wall that divided the two units on the first and second floors. Plans, permits and inspections were involved, though I don't know the details of what was inside their walls/ceiling. They had to strip it down to the studs anyway because there had been a fire in the house before they bought it and it sat without a roof for several months. It had been slated for demolition. You can imagine the mess.
Anyway, they ended up with two support columns (encased in attractive wood doric columns) supporting a beam on each floor where the wall had been. It actually looks quite lovely, open, but with an interesting architectural detail. Maybe something you could investigate...

I agree that you should at least have a structural engineer look at the project. In Boston, this runs $350 for a site review & report. Not a set of stamped drawings, mind you, but having someone with the skills to assess the structural health of your house for such an undertaking.

You could do it, sure, and it could look fine. However, it may not be readily apparent (and not for a few years) what loads have been improperly displaced...until the wall starts to separate from the ceiling (drastic, but could happen).

Having a professional engineer assess the situation is particularly important since balloon framing was in vogue during the period in which your house was constructed. This type of framing has a whole host of issues that most people aren't aware of, since the studs run the entire height of the house, instead of being attached at each floor level. Think how easy it is to push over a parallelogram shaped / / object. That's the challenge with balloon framing. Not something I'd want to go into lightly.

Good luck.

Balloon framing is not any less resistant to lateral loads than modern platform framing (it is probably a bit more resistant); both systems rely entirely on lateral loads being transferred through the floor planes to the walls that are parallel to the lateral (wind & earthquake) forces and these forces are then carried through the rigid plane of those walls to the foundation.

In any case, this is a row house with brick bearing walls.

In Boston an engineer's stamp would be required in order to get a building permit for this project and if the work was extensive enough, an investigation and evaluation report of the effect of the proposed work on the regulated systems would need to be submitted as described in Chapter 34 of the state building code.

I stand corrected. Macy is obviously your man on this one...particularly since I missed the masonry wall comments AND he knows chapter & verse of the MA building code. Macy, you are welcome to swing by & inspect my disaster of a house anytime!

I'm always willing to make a site visit for a fee but it doubles for anyone who calls me Macy.