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Learning phase here.

I'm contemplating a home made outdoor boiler for heating my home.

I gather I need a boiler (generally outdoors), a storage tank (generally in the basement) and radiators in the living spaces. In between is a variety of pumps and pex(?) piping.

Let's talk boilers... I have an "old" oil tank and an old electric water heater available. If I cut off parts that don't look like a boiler and stick them together, I've got the heart of the system. This would be installed in a "utility" shed, possibly sheetmetal or cement block for fireproofing.

Storage... I have an "old" oil tank but if it gets turned into a boiler, I have nothing... So I should scout for one or more tanks totaling ~ 500 - 1000 gallons, that would be superinsulated (build a box around them and stuff with blowin insulation?) This would go in the basement.

The radiators... in my experience there are two sorts. The "old school" steam radiators in my Grandmother's house and the baseboard "tube with fins" with a metal cover that I grew up with. But I sense there are other sorts. So let me describe the house and ask what sort I should be looking at for heating it.

The subject dwelling (just to sound scientifical) is ~125 years young three story and is split into three apartments. All three currently heat with pellet stoves. Total is a little over 4k sq feet living space + basement.

Two apartments on one side:

One apartment is pretty open (huge doorways between rooms) and bi-level. Quite well insulated. One pellet stove on it's first floor keeps it comfy except during extreme, extended cold like we had all this winter.

The second is also open format, shotgun (three rooms) and not so well insulated. In fact not actually insulated at all. One stove keeps it pretty warm despite the insulation, because the square footage is smallish.

The third apartment on the other side is more traditional in layout, three levels. Upper levels are insulated but the bottom level is not. One stove tried valiantly but ultimately was unsuccessful in keeping it comfortable. Plan is to add a second stove on the second floor, which should add enough heat to the equation for comfort.

All floors are hardwood. All 50 windows are double glazed replacements.

Two existing, massive, horrifically inefficient furnaces waft hot air into the house, or would if they did not cost the GNP of a small country to operate.

My half thought-through plan is to leave the existing furnaces in place, leave the pellet stoves in place and add this new system without disturbing the existing heating plant(s).

I assume...

I would plumb the radiators using (insulated?) pex from the basement storage to each of the rooms needing heat. The radiators are... those baseboard type? Something else? The areas would be zoned on individual distribution pumps controlled by thermostats (and an automation computer?). They would disable if the storage tank temp fell below... 120?

Oh yes, with all this storage water - another heat exchanger / circulator to heat the water in the existing water heaters.

I'm not 100% clear on the exchange of heat between the boiler and the storage. Is there a heat exchanger in both or just one? Does the storage tank send cold water to the boiler for heating or does the boiler send hot water to the storage for cooling?

Any suggestions especially on the radiators and circulator pumps / controls?

Any glaring mistakes or misunderstandings in my outline?

[BoilerHouse]

The following may be the simplest solution; Your house is currently forced air (I think) so you could install heat exchangers, which are basically kinda like an automotive radiator, in the duct work very close to the existing furnaces (i.e. the plenum) and use the furnaces fans to distribute heat through the house.  These fans would be hooked up to the existing thermostats. The water would flow constantly and the fans would cycle on/off to control temp.
Storage is optional.  Many systems do not have any additional storage.  Think of storage as a heat battery.  You send hot water directly into it and it is in constant circulation.  The thinking is that, with storage, the OWB cycles burn longer, hotter and is therefore a bit more efficient.  However, not everyone is sold on that idea.  If you do have storage, you are right in that the best place to have it would be in the house.  Any incidental losses would help heat your space.
As far as your choice of materials, shapes, dimensions and volumes would help with advice. Common dimensions seem to be; For outer shell - round, 60 inch dia x 6 feet long x 1/8 min thk.  For firebox, round, 36 inch dia, x 4 feet long x 1/4 inch thk.  Keeping in mind this is very general and there is lots a variability when building "DIY".