Is money made outside the state in which you live taxable by your home state. Example. I live in NC. I work remotely for a company based in NH. Is NC supposed to tax the money I earned from NH company

Yes, NC will tax the income you earned from a NH company. If your state of legal residence has an income tax, they will tax all of your income - even if it is earned in another state.

Your physical location while working determines where your earnings are taxed. NH should not tax this income, since you physically earned it in NC. If you are working remotely in NC for a company based in NH, you are considered to be earning that money in NC.

More often, this occurs when people live in one state (e.g. NC) and work in an adjacent state (e.g. SC). Unless there is a special agreement between those states, you would first pay tax where you earned the money (SC), and then claim a credit for that tax on the return filed in your state of residence (NC).