They don't really become indebted to domestic factories or corporations. During WW2, for example, the biggest lender to the US war effort was the US citizen via Liberty Bonds and other schemes to encourage private investment. During WW1 in the US, the government simply issued Treasury bills which were then bought by the Fed (if other buyers could not first be found). Historically speaking, central banks were often created with the express purpose of funding wars. The BoE and the Swedish Riksbank were both founded by private individuals who, in return for special rights (typically a monopoly on money creation in a given area), lent the crown money which it then used to pay for wars. Domestic central banks can still eat up a lot of debt to help finance wars, although they always run the risk of inflation as a result.

As the other commenter pointed out, if the state were to force factories to work without monetary compensation, then entire supply chains get disrupted and economic collapse would almost certainly follow. The solution to that is to either implement price controls on everything and become more self-reliant - this was essentially the German approach during WW2, with the obvious caveat that they were deploying slave labor and invading other countries for raw materials, which they'd otherwise need to purchase on the market - or to just borrow lots of money to pay for equipment from private suppliers. The US during WW2 had a mixed approach wherein there were price controls on raw materials vital to the war effort (aluminum, rubber, etc) and on some consumer goods which were seen as important to sustaining popular support for the war (and inflation control). The US price controls system was the OPA but IIRC every major WW2 power had some version of this.

Anyway modern states finance war by borrowing from their citizens, either directly (via war bonds) or indirectly (via inflation).

Forgot sources:

Tooze's "Wages of Destruction"

Tooze's "The Deluge"

Calomiris and Haber's "Fragile by Design"