Build a house without using glue?

By the time you reach your age, you should know that the best way to solve a problem is to not have it.  You should have also learned that you can have whatever you want, all it takes is money.  So, between the 2, you should be able to reach a compromise that meets your goals.  I would suggest you start with an architect who can control the material selection and construction specifications and contractors' contracts to meet your requirements.  For example you are eliminating all laminates.  And your requirement probably rolls over to paints and finishes, which might then go to synthetics.  Somewhere you will have to make a decision of what is acceptable and what is not, and what can be done and what can't.  And what you are willing to pay.

You are YOUNGER and couldn't build a small box with glue.

In real life, very little if any glue is used in home construction.

Dimensional materials refers to ANY materials sold in standardizes sizes, such a 2x4s. The term has nothing to do with how the materials are secured together.

You WILL use dimensional materials unless you mill your own lumber.

I've NEVER heard of anyone building a house with only glue.  Most use screws and nails.

sure.  all older homes were built without glue [construction adhesive].  Just requires more nails and other fasteners is all.  note that pre-manufactured roof trusses these days are all made with glue ... and they'll be in the attic so maybe that won't matter.  yes, an ordinary rough carpenter is supposed to be able to make roof trusses on the spot ... and i'd want an older guy who worked the trade back before pre-manufactured roof trusses existed and still remembers how to do a proper job with this.

Of course you can. 100 years ago houses had no glue in their construction if you did not want any. You just have to decide what you want to exclude. Things like drywall really don't have any glue and could be used if you choose. If there are places where glue might be helpful you could use hide glue. 

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Having said that, there are many modern products that use high tech glues that are not toxic so you should not shy away from using some modern glues. 

"Dimensional materials" just means using all sawn lumber, no engineered products like plywood or paneling or even glulams.

So you know, there are a lot of problems to solve in housebuilding that are difficult to do without resorting to engineered materials, but that doesn't mean your problem is insurmountable at all.  It just means that certain elements that have become commonplace may not be available to you if you're committed to a no-outgassing home.  So enormous window openings, double-width garage doors and the like may be off the menu.

What you wind up with is a house that has very traditional features: single width windows and doors, diagonal sheathing and the like.  NOTE: you will PAY and PAY HARD for these features.  Because the other thing engineered materials do is sharply reduce labor costs, and in housebuilding labor is a HUGE driver of the final cost.

NOTE ALSO: there are such things as site-built glulams, built with solvent free adhesives.  Believe it or not you can achieve nearly the same performance as a factory-made glulam, gluing it up yourself with dimensional lumber and plain old wood glue - yellow PVA adhesive, which is about as benign an adhesive as you can imagine.

NOTE ALSO ALSO: outgassing is difficult to eliminate no matter what you do.  Even plain wood outgasses for a few years after it's cut.  It'll be most notable shortly after milling of course, but it will continue to exhale for a good long time.  If your irritants go beyond the usual formaldehyde and aromatics, this could be an issue for you.

FINALLY: consider brick.  

PVA based glue is innocuous.

Oh, yes. Yes, it is. You may also want to use some bricks too.

The day of the stinky glue is nearly dead.  Besides when glues dry, they are no longer toxic.  The stinkiest being the Yellow Contact Cement. as is used for laminate or arborite countertops.  The carpenter does that work, building the kitchen cabinets and the vanity cabinets for the bathroom. a 1 day stink and it is gone.

 Even that glue has a waterbased competitor that does not stink and sticks like regular stinky contact cement.(made by the same company).

 Linoleum flooring has their glues and they are not eye watering stink either.

 Carpenters use carpenter glue which dissolves in  water while wet but is hard like plastic or surrounding wood when dry. and it don't stink.

I in fact CHALLENGE YOU TO FIND A STINKY GLUE these days.  It will be THAT HARD.

WHO KNOWS?  It may be the cure for the virus.  Painters who still work with oil based paints and lacquers don't get sick because the room they are in STINKS.  Then again it drives out the other trades so farther than 6 feet.  More like a day or two away.