I Need a Formula Text Field titled Eligibility with 2 possible options (Eligible and Ineligible).

Try this

IF(
[Are you a Resident] = "Yes"
and
([# of days since last service] >= 365 or IsNull([This is a duration field]))
and
[Are you at least 30 days behind on Rent] = "Yes"
and
[Age] > = 18, "Eligible", "Ineligible")