There's no legal minimum. There's a practical one, and it comes from fees and reserves rather than the purchase price.
Start with what the account has to carry. Custodian fees are largely flat, so a $40,000 account pays roughly what a $400,000 account pays for the same single asset. That's a much bigger percentage drag at your size. Then the account needs cash for property taxes, insurance, vacancy and repairs, all paid from the IRA and never from your pocket. If a $65,000 rural house needs a $9,000 roof and the account has $2,000 in it, there's no legal way for you to close the gap with personal money.
On the loan: an IRA can only borrow non-recourse, meaning no personal guarantee, and the income attributable to the borrowed portion becomes unrelated debt-financed income subject to UBIT, reported on Form 990-T. The $1,000 you read about is the deduction threshold, not the tax. Only the debt-financed share of net income counts, and depreciation and expenses reduce it, so a modestly leveraged rental often shows small numbers. Small isn't zero, and trust rates climb steeply, so somebody has to prepare that return, which is another cost.
At $40,000, buying all cash in a cheaper market or holding a fractional interest alongside another investor's account are the usual paths, and the fractional route brings its own set of rules about who owns what percentage. Some people also look at holding a mortgage note in the IRA instead, since interest is exempt from UBIT and there's no roof. Whichever direction you go, have a CPA who's actually filed a 990-T look at the numbers before you commit.
