There are a few issues you may need to address, I'm not an expert on large cartesians, however I do build deltas similar to this in size. While you are short of what I would consider large format printers, you are on the cusp of it, and things just aren't as easy as "I want bigger", in fact you will be surprised how quickly they spiral beyond what you expect. This list is by no means comprehensive either.
Your Z axis
Take a look at the CoreXZ printer by Nicholas Seward, it's similar to what you are describing.
I would change two things with your plan though, I would use aluminum extrusions (V-Slot!) for linear movement instead of heavy linear rails, and a bowden system for reduced mass/volume.
Electronics...
As mentioned, you are on the cusp of large format, which usually gravitate towards Nema 23's and large ($$$) stepper drivers. Efficiency is going to be paramount to wish to use Nema 17's, this means small pulleys for belts and slower speeds. You can run two steppers on one channel, however, your amperage is cut in half for each. The setup I mentioned above will handle common electronics much better. If it was me, I would be looking at either a Smoothieboard or Beagebone Black and a shield.
A few other things...
I don't know what nozzle size you plan to use, but if you plan on using a normal 0.35mm-0.5mm, you may wish to pick up a uninterrupted power supply, since prints this size can take a week! A power dip can wipe out a weeks worth of work and $100 worth of plastic. This is based on OUR printer, which will run absolute rings around what you are proposing. Your prints could take closer to a month.
Heated bed?
Think hard about what I just said about time frames, electricity and the smell of ABS being printed. That's going to run up your electric bill, especially if you vent to the outside, or need to run an air conditioner. If you do plan on a heated bed, this itself too will be problematic. You will need to source a heater (hint: China), and also a way to drive/control/power it. You will probably need 300-400 watts to power it, and this is beyond the amperage 3d printer controllers can handle. This means either a relay or external controller.
Depending on layer heights, you also will need a fairly modern computer to slice such objects. Our large delta (18in dia.x18in tall) has been known to crash most slicers while doing 0.18mm layers (much less 0.05!), and slicing at 0.1mm layer heights can take over an hour in Cura on an I7/16gigs ram/SSD equipped desktop. Slic3r, if it didn't crash, would take all day. Even after we helped find the bug in Cura related to this, we still crash it on occasion with large, not very complex prints. Even host software can be crashed trying to print large objects, a scaled up Tiki cup can bring down an older Intel Core based laptop running Repetier Host or Pronterface.
Lastly, what is your time frame/skill level/budget.
If you haven't built a Reprap based printer before, start now. Pick a design, build it, use it, learn it, THEN start on the big printer. Note that I don't mean go buy a Prusa kit to assemble, I mean buy printed parts and cobble it together yourself. It will pay for itself later. You will not only get a printer you can use to build your printer and better understanding of what's involved, but keep you from making some costly mistakes later. Not only that, but odds are, your printer will need a LOT of revisions before you get it working, much less happy with it, having a printer or machine shop is vital if you don't want to be driven to the poor house, as is experience.
If you have no experience other than using a printer, expect probably close to $2k by the time you are happy with it. If you have done a kit, but never done a Reprap from scratch, you might do it for 3/4 that. If you build a Reprap first, you could do it for probably $1000, maybe less. These are just guesstimate and could easily double or triple (especially if you are at the lower end of the skill chain).
If you have no printer experience other than reading, you could easily top $3k-$4k doing this project if you just dive right in, which unfortunately sounds like what you may be trying to do. Will you have a $4k printer? No. What tends to happen is you buy some hardware or electronics, then find it is just completely the wrong approach and then you have to go buy more/different parts and try another method. By the time you are done you spent $4k on a $1k printer. Whereas, had you built a $500 Reprap from plans, you could have two printers worth $1500 in parts and only having spent $1700. You will still make mistakes, but fewer of them, and less costly ones. This sounds extreme, but it will happen, your first printer you build/design always has cost over-runs. You try and cut corners where you shouldn't, use old/outdated ideas, use a product long deemed junk, break something, let the magic smoke out, etc. It happens.
Your Z axis
Take a look at the CoreXZ printer by Nicholas Seward, it's similar to what you are describing.
I would change two things with your plan though, I would use aluminum extrusions (V-Slot!) for linear movement instead of heavy linear rails, and a bowden system for reduced mass/volume.
Electronics...
As mentioned, you are on the cusp of large format, which usually gravitate towards Nema 23's and large ($$$) stepper drivers. Efficiency is going to be paramount to wish to use Nema 17's, this means small pulleys for belts and slower speeds. You can run two steppers on one channel, however, your amperage is cut in half for each. The setup I mentioned above will handle common electronics much better. If it was me, I would be looking at either a Smoothieboard or Beagebone Black and a shield.
A few other things...
I don't know what nozzle size you plan to use, but if you plan on using a normal 0.35mm-0.5mm, you may wish to pick up a uninterrupted power supply, since prints this size can take a week! A power dip can wipe out a weeks worth of work and $100 worth of plastic. This is based on OUR printer, which will run absolute rings around what you are proposing. Your prints could take closer to a month.
Heated bed?
Think hard about what I just said about time frames, electricity and the smell of ABS being printed. That's going to run up your electric bill, especially if you vent to the outside, or need to run an air conditioner. If you do plan on a heated bed, this itself too will be problematic. You will need to source a heater (hint: China), and also a way to drive/control/power it. You will probably need 300-400 watts to power it, and this is beyond the amperage 3d printer controllers can handle. This means either a relay or external controller.
Depending on layer heights, you also will need a fairly modern computer to slice such objects. Our large delta (18in dia.x18in tall) has been known to crash most slicers while doing 0.18mm layers (much less 0.05!), and slicing at 0.1mm layer heights can take over an hour in Cura on an I7/16gigs ram/SSD equipped desktop. Slic3r, if it didn't crash, would take all day. Even after we helped find the bug in Cura related to this, we still crash it on occasion with large, not very complex prints. Even host software can be crashed trying to print large objects, a scaled up Tiki cup can bring down an older Intel Core based laptop running Repetier Host or Pronterface.
Lastly, what is your time frame/skill level/budget.
If you haven't built a Reprap based printer before, start now. Pick a design, build it, use it, learn it, THEN start on the big printer. Note that I don't mean go buy a Prusa kit to assemble, I mean buy printed parts and cobble it together yourself. It will pay for itself later. You will not only get a printer you can use to build your printer and better understanding of what's involved, but keep you from making some costly mistakes later. Not only that, but odds are, your printer will need a LOT of revisions before you get it working, much less happy with it, having a printer or machine shop is vital if you don't want to be driven to the poor house, as is experience.
If you have no experience other than using a printer, expect probably close to $2k by the time you are happy with it. If you have done a kit, but never done a Reprap from scratch, you might do it for 3/4 that. If you build a Reprap first, you could do it for probably $1000, maybe less. These are just guesstimate and could easily double or triple (especially if you are at the lower end of the skill chain).
If you have no printer experience other than reading, you could easily top $3k-$4k doing this project if you just dive right in, which unfortunately sounds like what you may be trying to do. Will you have a $4k printer? No. What tends to happen is you buy some hardware or electronics, then find it is just completely the wrong approach and then you have to go buy more/different parts and try another method. By the time you are done you spent $4k on a $1k printer. Whereas, had you built a $500 Reprap from plans, you could have two printers worth $1500 in parts and only having spent $1700. You will still make mistakes, but fewer of them, and less costly ones. This sounds extreme, but it will happen, your first printer you build/design always has cost over-runs. You try and cut corners where you shouldn't, use old/outdated ideas, use a product long deemed junk, break something, let the magic smoke out, etc. It happens.