I am turning 65 years old next month, but I am not entitled to Medicare without having to pay a premium for Part A because I have not worked long enough to qualify. Can I sign up for a Marketplace plan?

Yes, in general, people age 65 or older who are not entitled to premium-free Medicare can purchase health insurance coverage in the Marketplace (except undocumented immigrants).  If you sign up for a Marketplace plan, you will be eligible for premium tax credits to make the coverage in the Marketplace more affordable if your income is between 100% and 400% of the federal poverty level ($12,140 to $48,560 for an individual in 2019).

Keep in mind that if you are able to continue working, you may be able to earn enough work history to qualify for premium-free Medicare in the future.  So another option for you to consider would be to sign up for Part A and Part B coverage when you turn 65 (you will have to pay a premium for both Part A and for Part B), and when you become eligible for premium-free Part A through your work history, you will then only have to pay a premium for Part B.