If you can find an IRA trustee to do it, it is allowed. But with strict limitations, including: Your IRA must own the real estate strictly as an investment, which means no use by you or certain family members. The IRA trustee must also hold legal title to the property. And finally, your IRA must have sufficient liquid assets (either from other investments in the account or from your annual contributions) to cover any costs associated with owning the real estate.

My guess is that the trustee, if you could find one that will do it, will charge a large annual fee to handle it.

Yes. (If the Apartment Complex is part of a publicly traded REIT)